A short guide to boarding (might get long...)
Boarding is a very worthwhile strategy for two reasons: It gives you more scrap/ressources and it's extremely reliable. You will find a teleporter somewhere, along with crew to use it. It's also extremely effective against the final boss if you know how.
There are two ways to defeat an enemy ship while boarding: Kill all their crew or destroy enough systems to bring their hull down. The first option is preferred because it gives more scrap and chances of crew survival are higher, and I'll focus on it.
To start boarding, you'll need a few things:
- A teleporter (duh.), level 2 is highly recommended because it allows escape from suffocation.
- Enough crew to effectively use it. 5 total is ok, but leaves one system unmanned. 6 is better, leaving you with a full crew on board. Mantis are the best fighters, followed by rockmen and possibly crystal, then slugs, then humans.
- A way to defend yourself - adequate shields, good dodge and anti-missile drones.
- A way to deal with enemy medbays/drone control if required - small bombs are best, breach bombs are ok, missiles work as well if there is defense drone.
- Sensors level 2 or a slug are extremely useful for figuring the best spot to land.
- These are the bare minimals, there are lots of other awesome tools at your disposal (healing bombs, fire bombs...)
What you want to do is to kill the enemy crew as fast as possible while taking the least damage possible. How?
If there is no medbay, teleport your crew into a 2-man room and start killing. If you can see where enemy crew is, try pointing them at a room with one person in it. That hopefully gives you enough advantage to kill at least two in one go. To distract other enemies, point your weapons at their weapon system - less damage for you, and another source of trouble for them. At least one enemy will try to keep it repaired, removing him from the fight.
Watch your health closely - once one of your guys drops below 25% health, you'll hear a beep and his health bar starts blinking. Hopefully your teleporter has charged by now... Once someone dips below 10 HP, beam them home.
Heal and repeat until they're dead.
With a med bay, things are marginally more difficult. Take the medbay out with your missiles/bombs, usually 1 point of damage is enough, rarely you need 2. Then beam your guys into the medbay - that way it can't be repaired because their dudes have to fight you. Careful though, those rooms may have room for 3 - you can be overwhelmed quickly that way unless you have a badass boarding crew.
When you beam out, take care that the medbay is still down before you teleport in again - you have bombs

Rinse and repeat.
Things to watch out for:
Cloaking
It can really, really mess you up. As long as they're cloaked, you can't teleport out. This means if you're running low on health, this might kill you. If they have cloaking, either take it out or tread very, very carefully.
Cloaking + Blast doors (grey doors) = very, very bad.
Blast doors
Grey doors means blast doors. That means you can't move around their ship, meaning you can't run away until you can teleport and you can't change the positions of your fighters. Tread carefully.
Drone Bay
...and no defense or combat drones active. That can mean two things: System repair drones, which you'll take out easily, and anti-boarding drones. I've not faced them yet, but they might screw you over. Try taking out the drone bay.
Useful stuff
- A Mantis kills a rockman, but dies to anything but an Engi afterwards (if he's at least a master). If a Mantis is at least level 2 in fighting, he can kill two humans/slugs and survive. Oh, and 4 Engi

Superior movement speed also means they can run away easily.
- Running away. If you're running low on health, you can order your crew to run around the enemy ship till your teleporter recharges. Doesn't work as well with Rockmen... And only if the enemy has no blast doors.
-
The Crew Shuffle. Not invented by me, and darthcaboose describes it a lot better

Basically, it's a way to switch positions with your fighters. Might not work with blast doors. Very useful to know, that way you can hopefully last longer in combat.
- If you have healing bombs, then you'll have an easy life. You usually don't have to teleport back and forth, instead you can just heal your crew back to full easily. Remember it has a cooldown though, and it requires a missile! I'm not sure if they can miss, for me they have always hit their mark so far.
- Fire can be an useful ally, but be careful - it can easily backfire if you are using anything but rockmen. It's one hell of a distraction though, and if you have Rockmen it's almost overpowered.
- If you're not sure how many dorks they have or if there are Mantis or Rockmen, think about softening them up a bit. Bomb the rooms they're in, set fire to the ship...
- Watch out for blast doors and cloaking. If you can, take them down. Blast controls have priority.
Oh yeah, this did get long. Hope I could help some at least
