What makes the game play faster is four things. Shorter turns of only 8 impulses. Simplified SSDs that don't have as many components as the Captain's Edition sheets. Simplified damage allocation rolls using only 1d6 per hit instead of 2d6 per hit. (That means you can just chuck as many dice as hits you've taken rather than rolling 2d6 a dozen times or so each turn.) And the option of playing without Energy Allocation. Doing that won't set well with purists, since Energy Allocation is as central to SFB as heat management is to Battletech, but if you get rid of it you'll drastically speed up play even if you'll lose some of the flavor of the game.
You might even want to even modify the game's order of play to make things go faster. For example, taking Donald Miller's house rules as a starting point I've come up with this.
- Each player rolls 2d6 for Initiative
- Movement (going from lowest initiative to highest)
- Non-weapon Activity (transporters, tractor beams, shields, etc.)
- Seeking Weapons Launch, those from previous round resolve
- Direct-Fire Weapons Fire (must fire Hell Bore before other weapons)
- Post Combat (Seeking Weapons hit, Explosions resolve, etc.)
- Damage Control
And even if you don't want to change the rules as drastically as that the Cadet's Handbook is fun. There's plenty of supporting material out there too. Jeremy Gray has a great page devoted to it as does the BoardGameGeek site. The point is that the Cadet version is a fun little game even if you're not a big fan of SFB's idiosyncratic Star Fleet Universe or of the complex SFB rules. Who knows? Maybe you'll like them so much you'll want to buy the Captain's Edition. In any case you'll probably want to pick up some of ADB's Starline 2400 miniatures to use in your Star Trek games.