Newb - Overwhelmed

Best place to start, read the wiki. It's getting more and more out of date since DA launched, but most of it is still really helpful. Don't be afraid to lose. The harder AI will trounce you the first couple times, but it's more rewarding when you DO learn how to beat them.

Play your first game on a mid-size map with a couple other random races. Set the AI to one of he easier settings. You're not trying to "win" this game, so much as figure out where everything is and how it works. As soon as you feel comfortable with the interface, start playing on Tough(Intelligent). This is where all the AI really hit's it's stride and your economies are equal. If you play too many games on the easy settings, you'll form bad habits that will hurt you on the harder settings.

1 turn = 1 week
52 weeks in a year (I think.)

Building ships is based on what tech you have researched. You need different weapon techs to equip weapons. Defensive techs for shields, armor, etc... Engine tech for faster ships. The available hull sizes are based on which ones you have researched also. The hull size techs branch of of Logistics (Medium Scale Construction, Large Scale Construction, etc...) As you unlock each class, you'll get a number of pre-built template ships in the ship designer. The template ships are fine for your first game, but building your own ships and keeping your designs updated is a very needed skill. You don't need to update your ships every time you research a new tech, just do it as the AI starts to build better ships then you.

do your own ship designs actually effect anything?

You own ship designs are usually much better than the default ships you unlock, because with miniturization and the later versions of weapons and defense techs (laser III or Armor II for example) are also smaller, letting you put more on a ship. Also you can custom design your ships to combat your opponents, for example if they are using beam weapons and shields you should use shields and mass drivers or missile tech. If you want a higher defense then you should play as a good aligned race, (just my opinion) when you research the proper tech (Xeno Ethics) you can choose your ethical alignment, Good races get a galactic wonder project that when built raises defense. I've heard a lot of people say that good is the weakest alignment, but I only started doing really well when I began to play good, (this is probably due to my play style more than any inherant qualities of the alignment) but don't take my advice to much to heart, you really have to just experiment and find what works for you. If you have any more questions don't hesitate to ask, good luck.

Yes, you'll be far more effective if you design your own ships using the best of all technologies available than you will trying to use the core ships, which have weak, fixed stats, and difficult to remember requirements.

Note that the AI designs its own ships as well using the techs it has. So again you will need to do the same to be competitive.

wow. I won a mission and I am not sure what I did. It was the first one in the campaign.

Do you have the Dark Avatar expansion? If so you should have AI use max CPU, its more difficult but many times more fun.

Yep. I have the Gold Edition. I played on the easiest setting.

the campaign is fun, but the real meat and potatoes of the game is the 'sandbox' match. this is also where you really learn to win. unlike the other games you mentioned having exposure to, most of this game isn't in the campaign. it's not bad by any means, but you'll see the most interesting things happen on the sandbox games.

Beginner's Strategy Guide This is a good place to start. It is written for Dread Lords, and not the Dark Avatar expansion pack, but the basics are still the same.

Welcome to teh game man, its much different than starcraft, etc. Some consider it a totally different genre.

One peice of advice that really helped me was "don't bend over a dollar to pick up a dime" I started out my games colonizing everything in sight. This is not a good idea. You have some time in teh begining of the game to build a small scout ship and explore the areas around you. Research the Stellar Cartography tech - it will allow planets to show up on your mini map. That way you can use your scout and colony ship to check out larger star systems that may have a good planet. The Class or Planet Quality (PQ) is really really important. Ignore those class 4 and 5 and go for the highest ones you can find first. The PQ determines how many building you can initally place on a planet (factories, lab, farms, markets). A High PQ planet is worth several Low PQ planets.

TBS *is* a completely different genre from RTS

Another thing you might try to do is colonize the planets furthest away from you and then work inward, thus limiting the number of planets your opponents can get, don't worry too much if they get a colony inside your borders, you'll probably be able to culture flip it later in the game.