Assault Kid of the Shuto Expressway

Defeat 10 Wanderers

Rookie of the Shuto Expressway

Defeat:

• B-001 ROLLING MASTER
  
• B-006 CRIMSON BOY
  
• B-010 ETERNAL POLARIS
  

Respective Speed Metrics:

• 278.70
  
• 298.84
  
• 282.41/361.55 (later battle as B-011)
  

Promising Street Racer

Defeat:

• B-003 BLOODHOUND
  
• B-007 GOLDEN HANNYA
  
• B-012 TRANCE DRIVE
  
• B-013 PLAYFUL GENTLEMAN
  

Respective Speed Metrics:

• 306.03
  
• 295.76
  
• 270.41
  
• 308.94
  

Emerging Street Racer

Defeat:

• B-016 CINDERELLA PAST MIDNIGHT
  

Speed Metric:

• 407.10
  

Shoot-Down Kid of the Shuto Expressway

Win 500 Rival Battles

Winning Streak

10 wins in a row

Battle traffic cars (a.k.a. The Nameless Ones / Other Cars).

Instant Kill

Win in <30 seconds

Battle traffic cars.

Rain Check

Draw 3 times

When leading, don't follow the rival's path.

Twists and Turns

Win after 10+ wall collisions

Battle a traffic car, do light wall taps on a straight.

PA Talk Lover

View 100 different conversations at Parking Areas

↑ These ones. On the map, a PA that has conversations that haven't yet been viewed will have a speech bubble with three dots above its icon.

Beautiful Cornerwork

Get CLEAN CORNERING in 5 consecutive turns

Battle a traffic car, drive as slowly as you need.

Strike First

Get FIRST ATTACK in 10 consecutive battles

Battle traffic cars.

The Art of the Comeback

Get REVERSAL ATTACK in 5 consecutive battles

Battle traffic cars. To perform this move, let the opponent pull ahead by more than 25 metres, then overtake them by more than 25 metres yourself.

Sharp Pedalwork

Get JUST ACCELERATE in 3 consecutive battles

Or, as the Japanese version calls it,
Battle traffic cars. To perform this move, do not step on the gas until a fraction of a second after "GO!" is displayed. Probably a mistranslation meant to say something like "Just-in-time Acceleration."

Hey! Taxi!

Win a battle with the Nameless One driving a Toyota Crown Athlete G Package (GRS184) taxi

While seemingly unwinnable at the very beginning of the playthrough, this battle becomes easy fairly soon into it.

Push It to the Limit

Push a car for 1 km

Locate any traffic car, get right behind it, step on the gas. Takes like 30 seconds. Honestly kinda pointless.

Car Enthusiast

Have 10 cars in your garage

This will probably happen in its own during a normal playthrough, but do keep an eye on your BP. There is currently not enough BP to be gained from defeating rivals to get every perk, so by not buying enough garage expansion upgrades and spending everything elsewhere you can lock yourself out of this achievement, unless you start a new game.

Full Time

Win a battle that lasted for 5+ minutes

Battle a traffic car, chill alongside it. Maybe avoid the C1, as the rival car can crash itself into losing on its tight turns over the required 5+ minutes.

Straight to the Bathroom

Enter a PA at 300+ km/h

This can easily be done at the Tatsumi PA, which is the easternmost one in the game. It's located right at the end of a massive straight, and there is barely any need to turn into it. Still, the required speed alone makes this a mid-game achievement.

Revenge of the Kei

Win 10 battles in a Kei car

Battle traffic cars in the Suzuki Alto Works (HA36S) you'll have to tune up for a certain rival fight anyway. W-012 WIND OF NAPOLI, Speed Metric 287.52, only battles against this exact car.

Slick Tires

Win a battle with tires that have 0 durability left

According to numerous reports, this achievement only counts against a previously unbeaten rival. So, if you've beaten them all prior to the achievements being added, you can either use a fresh save, which seems to wipe the old one — see below, or wait until the full release adds more rivals.

Thrill Seeker

Get 10 NEAR MISS in a row without hitting anything

Easy enough but could be fun for a short minute. You can drive as slowly as you like, near misses count even when done at <100 km/h.

Ginza's New Attraction

"Get stuck in the bridge pier in Ginza"

Badly mistranslated description. What it actually means is "hit a bridge support in Ginza while mid-battle." This bridge, located 1 km south of Edobashi Junction, will do nicely.

300 km/h Attempt!

Hit 300+ km/h on the Akasaka straight

The Akasaka straight is a stretch of the Shuto Expressway located within the Akasaka district of Tokyo. The district is shown in red on the map below.

According to numerous reports, the achievement is granted upon driving through the above-shown speedtrap at the required speed. Without using the save editor to give yourself overleveled parts, this appears to be extremely difficult or impossible at the moment. Hopefully, the full release will have more upgrades to install.

Back It Up!

Drive for 1 km in reverse at 100+ km/h

The idea is to increase your car's top speed in reverse, go to a nice long straight and drive backwards using the rear view camera. However, as you can see below, you can't individually tune the reverse gear's ratio. So, slide the final drive all the way to the left and hope the car will be able to achieve the necessary speed.

The levels of either the engine or the transmission don't appear to matter here, as the car will achieve the same speed with a GENUINE engine and a Lv. 1 transmission as it will with those parts at Lv. 6 and Lv. 3 respectively.
With that said, here's a table of some of the cars and speeds they were able to achieve in reverse with the final drive set to top speed (unless the car is Customized), so that you know which ones not to waste time with in case you don't have the ones that work:

LPS

Drive 1000+ km in 1 car

PA Encyclopedia

View 300 different conversations at Parking Areas

See "PA Talk Lover" above. You'll have to read every new conversation you come across for quite a while to get this.

Freewheeling Kid of the Shuto Expressway

Win 100 battles with traffic cars

Rake in Money

Get 300 million CP

This will be painful regardless, as a full playthrough gets you like 70 million. A good strategy would be to:

1. Level up the wallet
   
2. Enable the new "Water/Oil Temp" rules, found at the Garage screen, which multiply the CP received from battles by 1.5x.
   
   
3. Use the Lv. 3 Long Runner skill, which gets you increasing amounts of money from LONG RUN: 160000 CP at 25 km, 320000 at 50, 480000 at 75, 800000 at 100. With the new PERK system, prioritise skills that allow you to complete successive battles faster, such as Pride of Speed or Aura of Dignity. There is no longer a skill to increase rewards from LONG RUN, which are now: 140000 CP at 25 km, ? at 50, 420000 at 75, 700000 at 100.
   
4. Locate a high-paying rival, such as W-013 PERFECT REVOLUTION (Speed Metric 344.47)
   
5. Battle him repeatedly. The reward will rise with consecutive battles, up to over 200k 800k CP per race starting from the 21st consecutive victory.
   
   With all the bonuses counted, an ordinary ~30-minute ~50-win run will, therefore, net you about 40000000 CP.
   

*It used to make sense to do handbrake drifts right after the victory, getting on average some extra ~25000 CP, but with these new "Additional Rules" I'm not so sure. It's certainly easier to immediately stop and cool the engine as you wait for the rival to catch up to you.

The One with a Thousand Names

Get 100 different words from BAD names

I have no idea how one could get this many B.A.D. names without using a wiki. I have beaten every rival so far, and in doing so, have gotten only 17 different B.A.D. names. Here's a reddit post with a good list of available B.A.D. names (pointed out by IntroVert_Fresh_123 in the comment section).

Power of Continuation

Have 365 in-game days pass

Perhaps, the most ridiculous achievement. As there is only about 50 in-game days of good content right now, you can either:

1. Enter/exit the Shuto Expressway 300+ times just to get this achievement early, or
   
2. Wait and hope that there will actually be 365 days' worth of gameplay once TXR exits Early Access.
   

Personally, I'm going for the second option.