When you encounter the plague-infested station, if you have an advanced medbay and the Engi Med-Bot Dispersal augment, you can cure the entire station in less than a minute. Even the game notes that the colony leaders are astonished.
Having the appropriate weapons for the Giant Spiders: Gliding up to the airlock and letting an anti-personnel drone to quickly kill the creatures, or the more awesome (if less rewarded) option of slamming a boarding drone through the stations hull before it starts tearing the spiders apart, or the surgical option of using a bio-beam to pick them off one by one.
The Weapon Pre-Igniter plus a crazy-powerful weapon like the Glaive Beam.
If you're very, very lucky, you can get the Weapon Pre-Igniter as a post-battle reward, a surrender offer, or best of all, a random "found an item" event. This can happen in sector one. If that happens, have fun breaking the game.
For example, you can use the Pre-Igniter to defeat an enemy ship without letting it fire, or if the enemy is weak enough and your weapons are powerful enough, you can destroy it in a single Alpha Strike.
The first time you do the former situation will actually give you an achievement called "They never saw it coming".
Sending more than one Boarding Drone is also hilarious: punching holes in the enemy hull and watching your killer robots rampage through the enemy ship is awesome.
Automated Loader + lots of burst lasers + upgraded weapon crew member = fun for the whole family. Also note that the reloader can stack and is quite cheap.
You may encounter a Mantis Cruiser lined with Rock body parts. If you're flying either Rock Cruiser, you get an option called "Ram the bastards." Doing this will disable the Mantis ship's engines permanently allowing you to get guaranteed hits, in a very rare opportunity to preemptively disable an enemy's systems instead of the other way around.
Outwitting the Slugs, whether it be by using your own Slug's psychic powers to sense their attempts to plunder your ship, letting a Rock crewmember No Sell a spiked drink, getting a Mantis to slice them in half after they try to lure you off your ship, or just by forcing them to surrender after kicking the crap out of their ship.