This guide assumes using the first crew, but general practices should translate to others. Just the capabilities and priorities may shift depending on what you are lacking and what your starting capabilities are.

In general, you need to prioritise following:
Prevent crises from overwhelming you by filling all free ship spaces as the next one will end you.
Other than that, you need to keep basic ship states intact (>0):

• Ship status
  
• Oxygen
  
• Stability

Then you need to fulfil objectives to propel you further in the scenario progression. This is however the least priority of all, so do not rush it before you are well prepared.
But before any of preceding becomes real problem, your highest priority is to build up your deck. It takes back seat to very critical situations to solve, but always comes before progression and balancing it is at your discretion.

Note: After patch, % of rarities of cards in shop are objective progression dependent, so the priority goes slightly up - but still first have the basic AP gen. cards to buy the higher rarity cards.

In any case you need to build up Action Points (AP) generation ASAP.With the first crew you are in luck, it is relatively easy.

Captain Léon can generate 4 at cost of -5 stability. You can use it relatively freely as Jiao can regenerate it albeit at slower pace.

if you have either Léon or 4x consultation, you can buy immediately bronze 4 AP cost card from store. Or you can pass turn and get 1 AP per card in hand, usually leaving you with 5 AP for next round. Or if you have crew with support character (does not discard at end of turn and do not generate AP) 4 AP.

Alternatively, you can keep some cards in hand (right click, costs 1 AP and 1 Stability) to stack full hand to get 7 AP after next discard. You can also keep Léon or later any high yielding card while waiting for the right cards in store.

This makes relatively easy for going for silver cards from start, while super easy for bronze. With lucky hand, even gold is possible.

Now you have some AP so you are looking for following cards:

Bronze:

• Decision Making +2AP - rudimentary AP generation, better than basic consultation.
  
• Risky decision (50% +6/-3AP) - not reliable, but useful gamble early on. Use it after you deplete APs for only potential gain or gamble on getting some silver/gold in store.
  
• Xeno Amphetamines Cloning, + 5 AP, +5 Stability -2 oxygen. - Use only strategically to obtain high priority cards or solve most critical issues. They are cloning, so if you abuse them, they will flood your deck. Then get rid of them when you have opportunity, optionally keep just one or two until you have better options.

Silver:

• Battery +0 AP, gets +1 AP after play for 1 Ship status, max +7 AP after you play it 7 times
  
• Teamwork +1AP, draw 1 card - good after upgrade, it becomes welcome AP generation + deck cycler - good update candidate as it doubles the value.
  
• Negotiation +4 AP, discard random card. Keep in mind in what order to play this to minimise chances of disrupting your play by unlucky discard. Otherwise very good starter.

Notable omissions: Planning is possible with a discarding card, but unnecessary hassle with this crew. This is for crews with innate ability to discard and draw and no innate AP generation.

After getting the silver cards, you should have no issues going for golds and eventually the platinums.

Your starting crew has following capabilities:

• Léon +4 AP, -5 Stability
  
• Tina +4 Medical, draw card
  
• Jiao +4 Psychology, +3 Stability
  
• Jack +5 Combat
  
• Kapil +6 Mechanics, discards a card next turn

This means that you are relatively well covered, except one crucial element:

Authority

With this deck, I go straight to Militia as you can recover the lost Stability with Jiao.
Other possibility is Balance of Power which needs discard - either Ration or Encouragement. But this work better with some other crews, this one innate ability to deal with stability, some others may have innate ability to discard.
After patch, you have already Bullying It's basically a better Militia. Your priority is to upgrade it on the first opportunity by solving a crisis with upgrade.

Other skills:

Optional: Once you manage to get authority covered, it is very nice to get silver Mercenaries as if you keep authority high they can generate all skills necessary to either solve simple crises on their own or to complement your native skills for high difficulty crises.
Note: Now after patch it is a gold and therefore a late game strategy

Next one you need covered is backup Medical. It may happen that Tina gets sick and you need other medical card to resolve it. Or use authority if you keep it high. Bronze Antibiotics with 1 Authority can get you covered, or if you find silver Medical Team, the better.
Note: After patch you have 100 days where Tina cannot get sick so you only need it for harder medical crises and therefore priority is lower.

Now you are covered for most crises - if you keep authority high to compliment your crew. But it is costly to use a lot of it so next you want to obtain some backup cards for your skills.
You can now prioritise basic Tools (if you have not already obtained them) to solve first objective/crisis.
Note: If you have started with tutorial, you have it already solved by end of it. If not, solving it will give you more % of higher rarity cards after patch.
Then you can focus on psychology backup in case you get some cards hostage. Psychological support is generally useful silver card which is optional for this crew as it is a bit worse Jiao.

Then upgrade gradually on what you have and focus on deck cycling (draw), and deck.manipulation (exclusion, recruitment, cloning, perks etc).
I will not describe ultimate late play tricks. I will let you discover it on your own.

This should get you going.
There is way more to it but with these basics you should be able to build on and expand your gameplay.
I may add to this based on feedback on particular pain points.