General Advice: Play in a way that works for you. The Krynn are one of the easiest, most powerful factions to win with. Any style really works, because they are ideally suited for expansion, war, and influence. Pretty much any endgame outcome is available to them with ease.
With the Krynn, I played my normal way (Benevolent Ideology) because it's the best damn Ideology in the game, but also because it fits the Krynn's style of trying to spread influence and convert the other races. The Krynn have specialized buildings that play really nicely with Benevolent Ideology. The Krynniatic Temple has bonuses to Approval and Influence; you should be able to keep your coerced population happy and loyal (since you get coercion for free). The Krynniatic Ministry has an additional bonus to Tourism, so make use of that (Tourism is an awesome earner, especially on larger maps).
- Expand as fast as possible (first tech researches are engines for Moves so you can get to planets and resources faster)
- Research Diplomacy early for Embassies (buys time to build armies; keeps people from declaring war before you're really ready). Embassy bonuses play nice with Influence and Tourism, two things Krynn do well.
- Research Tourism. The AI will research trade. Trade for trade later. Tourism is lower risk (not subject to embargoes or war stoppage). Remember, you're the Krynn. You're not really interested in maintaining relations forever... Those trade routes will go away someday. Plus, the Krynn get some really good buildings for Influence/Tourism bonuses. Coupled with Benevolent Ideology (Missionary Center) and an Embassy, you can get a really nice 3 or 4 tile building set with a lot of bonuses. Setup several planets this way and you'll earn Ideology fast, you'll build influence, maybe flip a planet or two (or a lot when Ideology unlocks) and you'll keep people off your back for the first half of the game, until you decide to convert them with prejudice.
- Research Terraforming branch, Astmospheric Cleansing branch (to colonize such planets nearby), Hull Size, Logistics. Remember, the AI is going to research weapons; every race is going to research weapon tech. They won't research the specializations I want (shrinking), so I have to research that before I can trade for their specializations, but they will research weapons a lot. So I use my tech (terraforming, tourism, and others) to trade for a lot of the weapon tech. Saves me from having to research both things. What I don't do is trade them Hull Size or Logistics tech though; I won't trade away tech that makes war more difficult for me in a war. And if I research a weapon tech on my own, like Kinetic Weapons, I don't ever trade that stuff. It's mine. They have to figure it out themselves. No reason to give the enemy more power when it comes to war.
- Don't neglect research that improves Research or Production.
- Pay attention to Diplomacy. Keep up relations every 9 turns. Open Borders, that sort of thing. Keep people off your back. Pay attention - if a faction starts to dislike you, begin preparations for war. You need to anticipate what is going to happen and already be building ships before you need them.
Krynn have one of the best Abilties in the entire game: Coercive. They get the benefit of having a Bureau of Labor on every planet without actually having to waste a tile on the building itself. Use this to your advantage. Specialize every planet: Production + Wealth or Production + Research.
Of all the races, I enjoyed the Krynn the most and felt like it was the easiest win.
