I've read the strategy guides given here on how to start the game - max tech, get propulsion techs, spend to get 2 colony ships, then switch to max production and colony spam.
After colony rush with max production I'm far behind all other races in tech and warships (which you need tech for). I do trade with the AI factions for tech but still need to do extensive research. By the time my research is finally halfway decent the game stands at me having more colonies than any other single faction and usually second in tech (behind that blue-skinned race that always is #1). Before I get Planetary Invasion the other factions are declaring war on me, have a ton of ships, and invading my colonies. My ships have superior quality, but they have superior quantity and I end up having a stack of 10 ships blow up my star bases.
I've tried delaying tactics where I produce warships but station them around planets on the border of factions that are giving hostile messages ("You look tasty...") in order to stop invasion forces. That doesn't work because I usually have 3 factions declare war and there are too many planets to defend, and even with the defended planets that have 2 ships falls to a 6 stack and then there is a breach in the planetary line.
Can anyone give any suggestions on how to handle the crush of enemy warships and how to handle the mid-game?
Thanks
First of all, what kind of Galaxy settings are you plaing on? Size, Density, # Majors, Minors? Etc. On the smaller maps you typically have less "breathing room" since the AI's are much closer to your borders, resulting in them being in range (meaning you don't get a Relations benefit) and you will share borders with a lot of them (Relations penalty).
What I like to do, is assemble an early Military using the Resources available, e.g. Duranthium etc etc, in order to have as high a Military rating as possible. Futhermore, I will identify the most immediate threats (the ones most likely to declare war on me) and I will attempt to initiate trade routes with them, and possibly see if I can't start a war between that faction and another of similar power (if available). The trade routes provide a Relations bonus, whilst the war occupies the AI and makes them less likely to also declare war on you. Another benefit of getting Trade Routes going early on, is that they increase in value the longer they exist, meaning they can get quite productive. In my current game where I'm stomping, I'm getting around 5,500 Trade income every turn, without any of the mods that improve Trade.
Also, the Diplomacy tech tree gives Diplomacy bonuses, which are very handy for the Relations, and the Diplomacy giving buildings are (in my opinion) incredibly strong/broken against the AI.
I've tried to build multiple star bases - spam military ships. However only Earth has halfway decent production of medium sized warships (2 turns). Even other worlds that have 5 close sponsors and focused on factories can't produce as fast (4-7 turns each). This is probably because Earth has huge population + lots of tiles, and the outer colonies haven't had enough time to build the pop.
You have the right idea here, in that it is the population of your Capitol that enables it to have far superior production. It also has some extra "Raw Production" compared to your other planets, due to it being your Capitol. The way to improve your other colonies production would be to increase the population, just be careful to not make the planets too unhappy in the process 
