I was recently introduced to Eclipse via Big Daddy's iPad implementation. I've been mainly playing the AI as I try to get the hang of this relatively complex game. I'm getting better but I hope that veterans of the board game version can share some of their wisdom with us iOS newcomers.
While researching the below comments (but after posting the question) I found this excellent post, Five Thousand Words about Eclipse on the Tao of Gaming blog. It has some great thoughts about ship size, virtual fleets, and turn nine.
I thought I would answer with a few of my own observations regarding strategy to start off. Some of this answer is summaries of some articles on the BGG Eclipse Strategy Forum
As a general rule, only keep hexes in your territory if they provide at least two colonies. Territory with zero colonies (which you might have acquired to make a discovery) should be ditched immediately. Territory with one colony can be tolerated a short period of time (especially if they help you get to new territory). The easiest way to get rid of territory is to go bankrupt. i.e. take a few extra actions that will "force" you to remove influence the end of the round.
Plasma missiles seem to be the key to military victory. (At least in the base game, which is all that is currently available on iOS.) By mid game you must either be using them, or have a strategy to defend against them. I have no super successful defense, but improved hull and perhaps shield seems key: you need to be able to survive that first missile assault.
As a corollary to the above comment about plasma missiles, you should always have a consistent strategy for your ships. Don't invest in a lot of power generation or hull upgrades if you plan to take a missile based strategy. Don't duplicate techs (shields, computers, drives). If you think you will get the +3 later, don't buy the +2 now.
Be stingy with your turns. Avoid taking a turn to build if you don't have two things you can build. (Or three if you have an ability/race that lets you build three.) Same for upgrades: don't upgrade if you don't have two ship parts that need replacing. There are some exceptions, but turns are extremely valuable. Things that save turns or partial turns (like a ship discovery, tech discovery, or part discovery) or often worth taking.
Consider your turn order carefully. At the beginning of the round, contemplate how many turns you will likely be able to take. Consider your current money, how much money you are generating, how much territory you will likely control at the end of the turn (given expansion & combat), and any changes to those that will happen due to colonization/research. Perform actions that are likely to be "competitive" first, such as grabbing a newly available tech, or using a move to pin a ship, using a move to avoid being pinned. Upgrades/builds can often be deferred to later in a turn. (And can sometimes even benefit from being deferred since your opponents won't be able to react to them as effectively.)
Consider your ability to build a fleet, if needed. I've heard this referred to as your "virtual fleet". If you have spare minerals, spare turns, up to date ship designs, and available ship slots you can often build ships "as needed" to respond to an attack or to exploit a vulnerability. As such, I often avoid building my last few space stations/dreadnaughts except in response to an emergency. Conversely, when you attack, consider what your opponent will be able to bring to bear in response. An opponent with a developed fleet can be easier to attack than an opponent with a strong "virtual fleet".
The galactic center is very valuable. Not only do you get a discovery and a hex worth 4 VP, but the colonies themselves can turbocharge your production. The difference between a hex that produces 2 colonies and a hex that produces 4 (or more) is much more than double since you only have to pay the one influence. And, the galactic center also likely gives you roads to most or all of the other players. Which can be valuable diplomatically or militarily.
Some key "value" techs are: Neutron bombs (critical if you are going on offense), and improved hull.
Key "game changer" techs are plasma missiles (military game changer), techs that allow you to develop colonies (advanced labs etc.), advanced robots/quantum grid (saves huge money).
One quandary I still have is how much to "balance" your various resources. If human, I thought there might be benefit in pushing one as far as possible because the tracks aren't linear. i.e. if you get all of the way down minerals you can get 5 minerals per colony while the first few tracks of research are only a 1-2 research per colony. Why not try to push one all of the way and then just exchange at 2:1 if needed. In practice this seems to not work very well, however. Having one deliberately weak resource just causes to many problems.
Obviously exploit your racial advantages.