My largest problem is this. I expand quick enough (colonizer/carrier cap + LFs), cut off my enemy, toss up starbases, then I get bored. I economize like crazy, max fleet cap, and steam roll the enemy planets.
Do you? Really you can't ever expand too quickly (well maybe if abusing the TEC Rebel truce amongst rogues). Most good players have their cap go in one direction and their fleet going in another to colonize faster, sometimes building turrets so their ships can attack the next planet rather than destroying all the militia.
Instead of tossing up starbases and building a massive fleet behind them, what can I do after I've spread out quickly and encountered my enemies? (Fleet comp, tech, and so on.)
Yeah, the AI still have some issues with starbase. If you want a real challenging game, try not to use them at all. Typically I'll only use them so I can block off one AI to go for an all out attack on the other.
Perhaps you would find an early rush strategy more enjoyable. Most of the fast past MP games have people going for the kill at each others home worlds ASAP. The new players and the AI might find it difficult to repulse say a Sova carrier running around the edge of the gravity well using Empargo or a pair of Vulkoras desolators very early on. If done right you can completely cripple one enemy or even take them out if homeworld victory is on.
Another thing is rather than just sitting behind starbases while you build/tech up, nothing says you can go raiding while you prepare for the final blow. This is especially worth while to get your titan up, and light carriers with some bombers or some orgrov torpedo cruisers if TEC can do some damage before reinforcements arrive. And don't be afraid to take the long way around either if your opponents front lines are too heavily fortified. Also the AI tends to attack your worlds if you don't have a fleet present, so it can be a good way to get them to waste ships on your starbase, making them even weaker when you move to take them out.
What settings should I be playing on? (Team settings, game speeds, pirates, fleet size, yada yada.)
Its entirely on you. MP games are always on normal fleet, pirates off, all fastest, equal teams. However you're playing the AI, play like you want. Pirates may make the game more unpredictable, which may be a good thing in your case.