Basically at the beginning you'll want to scout everything you can reach. Send out the two scouts you start with and find a decent system for you colony ship. Meanwhile, build factories on the homeworld for a few turns, then make more scouts (say 5 or more). Early on, you don't have to worry about being attacked - just see what's around you, get a second world colonized and see if you can spot any alien ships. The extra scouts have two functions; first off, they'll speed up the exploration and secondly, you leave one over any good planet you find to drive off other people's scouts. Those won't be armed either, so they'll run off if you advance or even just sit there. Later on you'll have to worry about armed ships.
Don't worry about research at the very beginning. The main thing at that point is to explore, stake out a claim and set up some more colonies.
Once the homeworld can turn out a colony ship in about 7 turns, I'll build a few. Once a colony is established it's a good idea to send a few more colonists to speed things up - I usually send about 6-10. You'll want to keep some ships over the new colonies to drive off enemy scouts - if they haven't explored the world, they can't send troops to assault it. Unarmed scout will do at first, but a few (10-15) fighters will be better, especially if you happen to have close neighbors.
At this point, your main goal to grab everything you can lay your hands on. If you can cut off another empire from an area (by colonizing the good planets in their direction) then you don't need to rush on the planets farther from 'em. As to when to stop the first wave of colonization - it depends on the size of the map, number of opponents and how close the neighbors are.
Once you've got a few worlds, you'll want to have the homeworld finish building factories and throw some production into research. The main early stuff you'll want will be colonization tech (Barren World landing and the like), range increases and reduced pollution techs. At this point, your goal is to get some research done and build up your factories so the colonies are productive. As you run into other empires, try for trade agreements, see if you can trade any techs and later see if you can get non-aggression pacts. Mostly you want to keep people off your back. Early wars are a serious drain on resources that should be going to colonizing worlds and doing research. You'll need a fleet at some point, but it's best to keep it small if you can.
Getting into alliances can be a good thing, but it can also drag you into wars, so it's something to be careful about. You do want to make sure to have friends, though - among other things, it'll help out a lot once the council starts meeting.
Oh yeah, and if you get into a Final War, take off the kid gloves. Capture planets only to steal tech and to extend your range; bomb everything else out of existence. Hit the enemy with bio-weapons if you've got 'em and the enemy doesn't have the defenses (normally I don't use these because of the diplomatic effect, but once everyone hates you forever that really doesn't matter .....).