I offer advice different from that of Echelion.
1) Dock the flagship and relaunch it and redock it until you have done all four "corners" of the square containing your Homeworld. This maximizes what you can "see" at the start. Once in a while, a good planet is revealed or an anomaly or a mining resource for a mining base. For example, if you see a "green" one, shift tactics to get a constructor there before an AI does. (AIs start off with absolute knowledge of placement of mining anomalies and planets). Use that new knowledge to choose where to place the flagship when you give it its orders to "auto survey."
2) Dock and relaunch the starting colony ship in a chosen direction. The initial load is 100 colonists and you want it to have 250 when it founds a colony. Choose the direction to either be opposite the flagship or towards a juicy high PQ planet if one has been spotted already.
3) Create a colony ship design with an extra engine (if necessary, no life support modules), if your race setup allows this due to miniaturization and engine techs. Convert/Upgrade your mining ship to that design on Turn 1. Launch it the next turn in a third direction, unless the flagship has found a juicy high PQ planet already.
4) Especially at higher difficulty settings, a player has no time to waste early. I advise you to get both initial colony ships some distance from your homeworld before colonizing, unless a planet at least PQ 11 is found, and preferable greater than 11. Your goal here is to maximize the "envelope" of space in which your flagship can wander about discovering things, especially useful anomalies. A few 500 and 1000 BC finds are incredibly helpful in the early game. Those 1% bonuses to research and economy are also not to be sneered at.
5) After a few turns, get some planet(s) building more colony ships. You want to keep your homeworld population DOWN by syphoning off colonists to other worlds. This is to allow higher tax rates while staying at 100% morale for the population growth bonus. Your income is based on ALL population, not just those on the Homeworld. (This changes some later when you have good morale and revenue enhancing buildings on certain planets, but you will not have those techs and the income to build them for many turns.)
6) If you colonize a planet in a system that has another good planet in it, you can build a small hull colony ship on that new colony with no engines and get it to a near planet in a turn or two. Don't waste a good and expensive ship on it.
7) I don't use Trade, so I do not know about escorting freighters. I do research Trade early, though, to place the Economic capitol on the Homeworld and to trade Trade especially to Minor AIs so that they send freighters to me.