1v1 is usually a pretty small map.  I usually try to expand right into the enemy, because he's coming for me!

For TEC, you need a cap with some firepower, I think, (marza?) and cobalts early.  If there's time, I still like getting carriers out.

For Advent, small maps are tough.  The illuminator is still the best early-game ship for them (because the carriers are so expensive.)  But you need 3 military labs to get them.  I like to start with the mothership, and upgrade the colonizer ability as fast as I can.  That way, I can upgrade logistics on the first couple of worlds I capture.  (With a discount.)  I like to get 3 mil and 3 civ labs (for trade ports) if there's time, but on a small map, it's risky.

Vasari: there's a good discussion of them one or two posts back.  Can't remember the OP's name right now...

Bear in mind, I've only played a handful of 1v1's (on an appropriately sized map anyway), and this was before I got decent with advent (it was the first few games I played online).

Nonetheless, I too have found it useful to expand directly towards your enemy. In fact, since i am an aggressive style player, I usually do this on any appropriately sized map.

Obviously, very first thing you should build is at least 2 scouts. You always want to make sure you have at least 4 disciples to help with militia clearing, and if you don't go progen get a colony ship (if there are neutrals nearby, obviously make one as well).

One thing is for sure: you want to rush to illums as fast as physically possible. Whether defending or rushing, you WILL NEED THESE to fend against the inevitable LRMs that you will face. If there are fire or ice planets nearby, you might want to get up two civ labs first and get your colonize abilities first (and it's always possible to scrap them afterwards). Otherwise, it's actually very feasible to jump straight to illuminators (ie after buying initial fleet, do nothing but basic colonization, building three temples of hostility, researching illums, and massing them).

After getting illums, your absolute top priority is getting repair bays (especially if rushing) and trade ports up. Which comes first depends on your situation. From there you will develop counters in accordance to what your opponent is creating, though personally I will usually always get a few drone hosts out just to ensure I have fighter cover for my illums. If they pump out a bunch of flak, you can always switch them to bombers.

As for starting capitals, there are strong merits for Progen, Radiance, and halcyon. It just depends on your strategy and preference. I can see some possibilities in the revelation, but I would wager that you would have to be particularly skilled to pull it off. NEVER start with the Rapture. If you do I will laugh at you with the levity of a thousand clowns.

If you found my post helpful, please give me some Karma I currently don't have any 

In a 1v1 (presuming single-star; if it's multi-star it's another story) going econ at the start of the game is suicide.  You're basically gambling that the enemy isn't going to scout you and attack you before your econ pays off.

As far as economics go, the best investment that exists is actually colonizing new worlds.  If you spend your money building trade ports between your home planet and two asteroids, and I spend my money colonizing two extra desert planets, my income will be higher and I will have way more logistics slots and I didn't need to pay a cent for civic labs and my military is probably stronger than your's.

Econ rush only works in team games, multi-star maps, or against incompetent opponents such as AI's.  I would never advise attempting it in a 1v1 scenario against a real player.

Early on, long range frigates (assailants, LRM's, and illuminators) offer the best balance of damage, cost, and health out of any unit available, and should probably make up the bulk of your fleet.  Depending on your faction, having some carrier cruisers, support cruisers, and flak frigates would be great additions.

Mid-late game, you should be slowly phasing in more carriers with bombers, support cruisers, and heavy cruisers. Remember, heavy cruisers are only countered by bombers, and bombers are only countered by fighters.  If you have strike craft supremacy, the enemy has to rely on special capital ship abilities to do anything at all.

Malice is a good skill, but shield regeneration is definitely the better choice. This is especially true once you have guardians.  Guardians + shield regen is easily the most powerful combo in the game.

Bring cap ships, but be a minimalist about it.  Two or three high level cap ships are better than five or six low level ones, mainly because you're paying A LOT more for the quantity.

Haha I literally laughed out loud.

Thanks for the karma, much appreciated