Approximately when do you upgrade to the next fleet level, and when do you get your second capital ship? After you have two capital ships do you split your fleet or do you still combine it?
I will usually only increase fleet supply when I need more, never sooner. The first upgrade should be done ONLY when you've reached the limit and need more. It's difficult to say exactly when you need to upgrade, but one should always consider one's economy when deciding. The first three fleet logistics levels can be purchased without economy upgrades as long as you have enough planets to support. Beyond that you will need economic upgrades like trade ports, extractor upgrades and culture beacons to prop up your economy. The best thing to do is to keep an eye on what your enemy is building. Research and build a few of the counters to whatever your enemy is building until your un out of available fleet supply. Then when you've hit the wall, start upgrading your economy and your ships stats-Hull/shields/Armor/weapons/abilities. When it comes to the point where you need a new counter unit or significantly more of a certain counter unit, it's time to upgrade fleet supply and build more units. Its Ok to be a little behind your enemy when it comes to fleet size, so long as you have the economy to support yourself if you take some hits and your ships are as strong or stronger than your opponents. Remember that repair bays are a defending fleets best friends, the more the better.
As for research and build orders beyond the first 10 minutes, not many can tell you exactly what to do because it depends so largely on your opponents and your surroundings. For example- Advent have cheap upgrades for planet population on Desert Planets, which is great if you have a couple of desert planets nearby you plan on colonizing. If not, than its not necessary to purchase these upgrades. There are some things that are always useful to research like terran population increase, health/armor/shields, structure build rate increases etc.
What I CAN tell you is what ship upgrades you'll want to get first. For TEC you'll want to put your money in Health and Armor, and health especially if you're against a Vasari opponent. For weapons, you'll need to decide whether you're going to stick with LRMs as your main damage dealer for awhile (Miissiles) or you're going to tech to HCs(ferro-uranite rounds) pretty quickly. For ADVENT go Health (especially against Vasari), Shields then Beams. Advent have really weak hulls so you definitely want to upgrade their health to compensate, also put some points into Knowledge Aggregation if you plan on doing a lot of research, especially on ship boosting tech as it will speed up research considerably. For VASARI I usually go Health+Reintegration and Phase Missiles. It's nice to have reintegration ready by the time you bring out HCs which will use it frequently to stay alive, and Phase Missiles are on Assailants, Flak, most cap ships and all your strikecraft, and will deal more and more damage straight to your enemies hulls the higher you upgrade it.
As for your second Cap ship, there are a few 'milestones' I look for when deciding to construct a second. Here are some of the indicators I use:
1. If my fleet has defeated my nearest adversary's at their HW and my first Cap ship begins bombing the surface.
2. I see an enemy Marza reach level 5. (need those ability interrupts against Missile Barrage)
3. If I've reached the 4th fleet supply upgrade and don't have one yet.
4. If I've got the spare resources and fleet supply lying around.
If there are planets nearby that you havent colonized, send your second cap ship with a couple of lrms (say 5-10) and colony ships to grab them (unless of course you need the seconds' abilities on the frontlines ASAP) If there isn't any uncolonized worlds nearby, include the ship in your main fleet or construct a secondary assault fleet so you can hit two places at once. Also, once your fleet reaches a reasonable size, attack the pirate base to gain a lot of XP for your caps, exclude any caps you have that are already Level 6 or higher.
Certain offensive opening strategies can benefit by building 2 cap ships before building a fleet (sova rush for example). But you should note that you will be unable to defend yourself against attackers for a dangerous period of time. If you DO decide you'd like to try a Cap Ship rush, you've got to make sure you know EXACTLY what ships and techs you will need and when to build/research them, and you'll need to make sure your attack is deadly if your opponent does not defend himself promptly.