Your starting strategy is pretty much standard, except for a few things.
Getting Universal Translator that early is usually not that important for most races. You might be better off going for Planetary Improvements (or its equivalent) to get better infrastructure (labs, factories, markets, etc.) or Space Weapons to build an early army. Preferably a mix of both.
However, for the Terrans and Drath, going for UT first is actually pretty smart, if you are able to quickly make contact with other races. The Terrans and Drath are the best diplomats in the game, especially the Terrans, so you can get very good deals while playing them. Buying and reselling techs from the other races can get you a pretty good profit. Just keep an eye on how much money your trading partner has. The less money they have, the less they are willing to pay.
Your starting colony ship is best used to find another planet. The secondary planet of you home-system isn't worth your time at the beginning. Also, if that planet gets colonised by another race, it is very likely to flip to your side, because of your influence. Oh, and don't forget to restock your starting colony ship before sending it out. It only starts out with 100m colonists, despite being able to carry 250m.
Using the social focus isn't that useful, if you are doing it with all of your worlds. Instead, use the production sliders in the Finance Management screen. Set Military Production to 1%, and split the rest between Social Production and Research. If one of your colonies isn't constructing any improvements, the money will be automatically allocated to ship-building. This has the same effect as what you were doing, but is less wasteful.
As for what you can do after the colony-rush? Well, there are a few things.
First off, research Trading, then build some freighters and send them to your neighbours to establish some trade-routes. Trade-routes not only increase your income, but also your relations with the races your trading with. You can also build some Economy starbases and fit them out with trade modules to increase your income even further.
Building starbases in general is something you can and should do. Economy starbases can not only be used to increase your trade-income, but also to improve production on your world.
Influencer starbases can be used to spread your culture and eventually flip planets of the other races to your side. However, you need to have a military, as that could likely lead to war. The other races don't take kindly to cultural conquest for some reason.
Military starbases are used to increase the fighting power of your ships. They do this via modules, that either increase the damage, defense or speed of your ships. For the first two types to work, however, your ships need at least one weapon or defense.
New starbase modules are unlocked by research, but to add them to a starbase, you need to use another constructor ship. This means, that building up starbases takes a lot of time and money, but it can be worth it.
Also, while the first few starbases are free (the amount depends on map-size), all subsequent ones cost an increasing amount of money to build. The cost can be reduced by researching techs that increase your Logistics ability, but it will still reach an exorbitant amount, if you build lots of starbases. However, there is one exception. Should your Logistics ability ever reach 100 or higher, all starbases will be free.
As for research, it depends entirely on what you want to achieve. Having at least a token army to prevent the other races from picking on you is mandatory. Besides that, you're pretty much free to choose. I usually go for better infrastructure first, because it improves the total output in research, production and, most importantly, taxes. However, if you are playing the Terrans, you can get those techs quite cheaply from the other races, so it might be better to research techs that improve your Diplomacy ability first, as that makes it easier to get even better deals. If you're lucky, you might be able to buy a planet or two.