I'll assume you're playing single-player only at this point.
Start by playing on a small map; either a random generated one or something like Point Blank.
Play as TEC.
On a small map you will only have one or two computer opponets. Set them to Easy or whatever the lowest difficulty setting is.
Opening strat generally, for me anyways is:
1) Build Crystal extractors
2) Build Capital Ship factory
3) Queue up a capital ship (usually a Kol or Marza) You can queue up a cap ship as soon as you've told a builder to start building it; you don't have to wait until it's done being built or even until it's started being built.
4) Buy 200 crystal from the black market. Just click on the purple Buy Crystal button in the upper right of your screen twice.
5) Upgrade your Planet Infrastructure if it's not maxed out.
6) Build Metal extractors
7) Build a scout; when it's done being built send it to the nearest planet and then set it to auto explore
8) Build 2 or 3 Cobalt Light Frigates
9) Build a couple of the Civics Logistics labs (I usually do this only because I want to research either the Volcanic or Ice planet colonization so that I can readily colonize one of those if they are nearby). Otherwise build Military logistics labs.
10) Once the cap ship is ready to go, send it along with your 2-3 Cobalts to the nearest easily attackable planet. Meaning the one that only has a couple of neutral faction ships in it.
Then just play around. See what different stuff does. At the beginning don't worry about micro-managing what your fleet is doing during a battle; just let them auto attack. Don't worry about what ship counters another ship. Worry about that stuff later if you want. Upgrade your cap ship however you want with whatever makes sense. Let the constructors place Logistics and Tactical structures automatically. Later you can figure out where good placements for them are but on Easy difficulty it just doesn't matter.
This is truly an easy game to get into and figure out without ever reading the manual or going through tutorials. Pay attention to the tooltips. They have enough detail to help you without being too wordy.
After a couple or three games on a small map, move up to something like Razor's Edge, which is in the Medium maps area. That's a fun one where you team up with two other AI teams and battle against 3 AI teams (I would suggest Locking the teams when you first play a team game).
Keep playing as TEC until you feel like you have a good grasp on the game and on how each ship functions, then go play as Advent or Vasari. They are basically the same with some variations here and there. Make sure that at some point you play a game where you max out the Civics and Military research trees, just so you get a feel for what each thing does.
Have fun.