The thing to note about Spacedock is that it is presented as a reference book so that you can easily find the bits you need in a hurry.
That's why it begins with ship construction, but if you're just going to use the rules I would actually suggest you skip that for a while until you get a good grasp on using Spacedock rules. After all, Steve already made Spacedock versions of most ships you might want in the Ship Recognition Manuals, and since they're all free for download you might as well get them.
So just skip Chapters 1 (ship classifications) and 2 (starship construction) and go straight to Chapter 3 on Starship Combat - that's the chapter you want to begin with.
You want want to note the comments on initiative on p.82, but else you can go on to the 'Character Roles in Starship Combat' bit on p.84 which is fairly important in describing what crewmembers do on a starship during battles.
Power allocation is the thing to keep most track of (p.94) except for combat itself. The conn/pilot should note the maneuver tables on p.103-107 and pay attention to the differences between attack and defense maneuvers. For use of weapons and damage, the Hit Location tables on p.112-117 are the most significant. Damage done is in SU (Structure Units) as listed in Chapter 2 (Starship construction). Also pay attention to shields - it's pretty easy to forget the importance of Shield Threshold and so.
Beyond that it depends on the Narrator. For example, I use the Starship Tactics skill to determine initiative very much like the rules say when I'm the narrator. My own narrator, however, likes to use the Command skill instead, whereas I just allow the command skill to slight influence the Test results of the bridge crewmembers in general (unlike Starship Tactics which influences only the Test result of the tactical officer)...
"We think we've come so far... Torture of heretics, burning of witches - it's all ancient history. Then, before you can blink an eye, suddenly it threatens to start all over again..."
- Captain Picard, "The Drumhead" (TNG).