1. Time
You have 60* seasons (turns) to achieve manned moon landing, but time is not on your side really. You won't achieve much during first 2 budget reviews (roughly 30 turns), although it's very important time when you recruit and train your SETs, controllers, crews. Then you will have to launch missions almost each turn in order to remove penalties and finally get smooth moon landing.

*I think to get achievement you have time unil Q4 1969. Someone please correct me if I'm wrong.

2. Money and prestige
Money is tight right until moon landing, so you can't waste it. At the same time you need to get just enough prestige for max budget and maybe occasional random budget increase when missions are successful. First 2 reviews are crucial to get enough money for harder and more important programs and rockets.

Here go DOs and DON'Ts

1. everything costs an upkeep, so don't just spend money, because you can, in particular:
   
   • don't open programs until you have team free to research components
     
   • don't upgrade buildings just until you need the room they provide
     
   • go cheap, I mean it, look how much the rocket you choose costs per launch and how many missions it can serve, e.g. the cheapest R7 Sputnik rocket not only can get you 2 satellites into LEO, but also uncrewed Vostok suborbital flight. There are other benefits of going cheap I'll explain in research part.
     
   • when training choose wisely, you don't get refunds when you cancel and train something else, even in the same turn!
2. plan ahead to the season, e.g. start upgrading VAB, so it ends just when your team is becoming free to open higher tier rocket program
   
3. close down not needed or finished programs, also try to keep them open as shortly as possible to finish, they take a slot and cost an upkeep
   
4. remember that each launched mission costs, so don't fly them until good reliability - perfectly 90+%, but 85+% or even lower is acceptable for uncrewed missions, depending on launch cost
   
5. don't bother with costly programs (in terms of money and RD difficulty) that don't lead to your goal, e.g. PKA space plane, where cost per launch is attrocious

You can get enough prestige for max budgets even when choosing cheap and easy programs.
You can, or rather have to, run small deficit during first budget window, I went -650 per season and survived, but you mustn't run out of money before review, so do the maths.

3. The crews
The only attribute that matters is learning potential, you don't want anyone with <70 skill to do anything, but training. Perfectly you want to train everyone to 80s, and controllers even to 90s, but use 70+ skills if time is crucial, e.g. you lack prestige before review. Each training session take 3 seasons, plan accordingly. Training gains diminish the higher the skill is and the lower learning potential.
Morale - you keep it high by making them busy, with training most of the time. If you have your employee already trained, but morale is getting under 65, train him anyway, so he doesn't quit. Or give whole team a rise, which may be more costly for larger groups.
On the other hand when average team morale is about 90 with the lowest one of about 80, you can cut the wages and get some relieve on budget. Most of the time you can get away with cuts on cosmonauts, because they train constantly and tend to have high morale. Too bad they have the most dangerous job too.

1. SET crew
   
   • use teams of 4 with the same skill for research
     
   • 20 people is enough for a game, you want by rough count 12 in first budget window, 20 in second
     
   • you need 2 rocket teams (hrr and rocket); when they don't work on rocket programs, it's best to train rocket team in EVO suits and hrr team in crewed spacecraft - ymmv; by the time you research those you should have all needed rocket programs researched already
     
   • you also need 1 space probe team, and 2 crewed spacecraft teams (1 of them could also be trained in space probes if you have time)
2. cosmonauts (Russians use cosmonauts, Americans use astronauts, but does it really matter?)
   
   • 5 is enough for a game, but you want to fill 7 slots eventually just in case of accidents
     
   • training 1st tier - fitness, leadership, piloting; 2nd tier - EVA, science - although I don't think I've ever seen EVA in mission tooltips, even spacewalk, and I looked hard. Tooltips bug sometimes though and show previous mission requirements instead.
3. controllers
   
   • the final mission (and few before that too) takes 15(!) controllers, and you must have them trained before
     
   • you need 10 controllers trained asap, 13 at the start of second budget window - no mission before Soyuz 7k L3 needs more, 15 as soon as budget allows. You may go for 17 depending on budget and use 2 additional just for director training, but I didn't think of it before my game and don't know if there's enough time for that.
     
   • final mission setup: 1 propulsion, 5 trajectory, 3 systems, 2 payloads, 2 ops, 2 directors
     
   • Vostok program (which I recommend to run first) requires 1 less trajectory controller and 1 more systems controller, so train 1 in both skills, systems first
     
   • probe programs don't need ops controller, they may be trained reasonably late, still one is needed for Vostok program

Year 1955 example with explanations:
Q1

• buy cosmonaut and mission control centers - training takes time you don't have so you need to start asap
  
• hire 2 SETs with highest learning potential - anyone you hire should have 86+ learning potential, don't bother with lower as it takes forever to train them already
  
• train SETs you have: 4 - rockets, 1 - space probes. You could go 3 and 2 if starting SETs have terrible rocket skills.
  
• there's no need to do anything else - you can't start researching until your SETs are trained, why pay the upkeep?

Q2

• hire 5 controllers - highest learning potential
  
• if you were lucky and your SETs would go 70+ in rockets after single training you can start building VAB now, but I think it's still better to train them once more instead and postpone VAB, so it's built just when second training session ends. That way you spare the upkeep and have quicker research later

Q3

• hire 4 cosmonauts - highest learning potential, but age matters a bit here, you don't want to train cosmonaut just to retire before moon landing
  
• train 2 new SETs in space probes (or 1 and 1 if you went 3 and 2 before)
  
• upgrade SET center, so you can hire new crew in 1956 Q1 - you want at least 12 of them in first budget window, and 20 at the beginning of second budget window

Q4

• 5 SETs finished training session, send them to train again or start researching rocket and satellite programs if you built VAB
  
• 5 controllers are ready for training, you need one with highest average skills (also highest learning potential) as director - train the weakest skill at a time, other 4 train: propulsion, trajectory, systems, payload
  
• upgrade control center, so you can hire new crew in 1956 Q1 - you want 10 of them training in first budget window, 13 at the beginning of second budget window

1. Research

• Pro tip - the most research is done indirectly. How does it work?
  Basically when you research a rocket or a component (R/C), a bit of reliability cascades down the line and increases reliability of other R/C up to a maximum of 75%. Thus you want to start your next research program as close to 75% as possible. That is why I stated earlier that going cheap is teh way. I started with cheapest R7 Sputnik booster at 3% reliability, but my next R7 Luna booster had over 20% reliability by the time I started research. This cascaded further, so when I finally got to N1 booster research, it was already at 75% reliability. Components have much lower gains, at least for the path I've chosen.
  
• Use 4 man research teams to get the most gains.
  
• Your research team should have higher skill than RD difficulty, it doesn't matter much at first, but later programs are much harder to research, so it's better to train your team (or just those with lower skill) to required level beforehand. That is main reason why the only viable Soviet moon landing program is L3 Soyuz, the other one has 100% RD difficulty and research takes forever even starting at 75% reliability, not to mention cost per mission.
  
• The higher the reliability the lower gains, but you can get as much as 10% (that I've seen) from actual missions, so you should think of starting missions when required R/C have reliability at 80+% especially when potential prestige and money loss is low. The less steps mission has and the better your controllers are the more success probability. That said it's usually not necessary to rush things and research is safer bet, there's just enough time.

2. Programs
TL;DR just the list I used and think is the most efficient cost- and research-wise
- boosters in order

1. R7 Sputnik
2. R7 Luna and R7 Vostok
3. Proton K and Proton KD
4. N1

- programs in order

1. Sputnik 1
2. Sputnik 2
3. Luna 3
4. Luna 2 and Korabl
5. Vostok
6. Zond 5
7. Luna 15 and Soyuz 7K-L3
8. L3 Soyuz

Longer version:
- boosters

1. R7 Sputnik - cheap and fairly reliable - used for: Sputnik 1 and 2, Vostok uncrewed suborbital - you can wait for R7 Vostok a bit longer, thus you can train SETs a little longer to have better research later, also spares upkeep early when money are most tight
   
2. 2 rockets
   • R7 Luna - cheap and fairly reliable - useful for: Luna 3 and 2, also Soyuz 7K-L3 uncrewed suborbital, but upkeep cost prevents waiting for it, also not enough reliability gained from just 2 missions
     
   • R7 Vostok - cheap, reliable - useful for: Vostok program, Korabl satellite, also Soyuz 7K-L3 suborbital missions - that's a good way to increase Sojuz 7K-LOK reliability, while using tested (reliable) and cheap rocket
3. 2 rockets
   • Proton K - fairly low RD difficulty, boosts reliability of N1 down the line - useful for: Zond 5, Luna 15, potentially Venus/Mars missions if time is available
     
   • Proton KD - fairly high cost and RD difficulty, but boosts N1 reliability - useful for: Soyuz 7K-L3 LEO missions - cheaper to use and available far earlier than N1
4. N1 - required for lunar landing, no need to argue

- programs

1. Sputnik 1 - cheap, easy (thus quick), boosts Sputnik 2 reliability, can go for extended mission for additional prestige and controller training
   
2. Sputnik 2 - easy enough for quick early prestige - open when Sputnik 1 research is done
   
3. Luna 3 - required to open other lunar flights - open when Sputnik 2 research is done
   
4. 2 opened at about the same time
   • Luna 2 - uses the same rocket as Luna 3, so it would be a waste not to use it, fairly easy prestige gain - open when Luna 3 research is done (see the pattern? ;-) )
     
   • Korabl - uses Vostok spacecraft - open when crewed spacecraft team is ready and able - cheaper than Vostok program, uses the same equipment
5. Vostok - your bread and butter Man in Space program, cheap, reliable, gets required goals done - open only when Vostok spacecraft and R7 Vostok booster are ready for missions
   Now, Voshkod program would be fair alternative, but it's more expensive, takes more research time (rocket and components), requires EVA research when you don't have your team ready yet, it would boost lunar program EVA reliability, except not worth the hassle overall.
   
6. Zond 5 - optional, but good for prestige boost, requires Proton K rocket - open when space probes team is free
   
7. 2 opened at about the same time
   • Luna 15 - optional, good for goals (I don't know if soft lunar landing experience relates to anything in next lunar missions, probably not) and prestige - open when space probes team is free
     
   • Soyuz 7K-L3 - huge cost, required to reach some goals and remove penalties before lunar landing missions, also good to boost reliability of components using cheaper rocket, while waiting for N1 to be ready
8. L3 Soyuz - lower cost as it uses the same components as above program; required for lunar landing, you can go for lunar flights quite early as only command module is required for those - open when N1 booster is fairly reliable to fly with

Additonal comments:
When possible try to use tested (and cheaper) rocket with not so reliable (yet) component in suborbital uncrewed flights (easiest).
Same components used in another program cost nothing (for program opening) - one of the reasons Vostok program is so good.
I kept my space probes team busy the whole game, there're some fairly easy missions to keep prestige going, while you're preparing for harder missions.
Don't even think of Lk700 direct ascent program, not for this achievement. I would be amazed if anyone pulled it off without save/reload spam before Q1 1970. Doesn't mean it's impossible, just improbable imho, good challenge though.
As for save/reload - the game probably generates seed every turn at the start, so one would have to reload 2 turns before failed mission and repeat all steps. It's far too tedious and completely unnecessary.
Tiger-teams - do they work? are they worth it? I very much doubt it, for them to work the genereated number would have to fall between: no tiger team option and chosen tiger team option percentage. Unless I'm wrong and it works quite differently than how I think it does.
I treat them as a talisman and use sparringly to get success chance to 90% if I have funds. I'm more inclined to do so when manned mission is at stake (it's irrational, but I'm just a man), especially when it's close to last stage (♥♥♥♥♥♥♥♥♥♥, failed recovery from the asian steppes).