How many units of people should remain in the first settlement? Is it better to create more settlements very fast?
My personal strategy is quite specifically to remove the tools from the initial pioneer. Sell them to a nearby Indian village.
Then use this money in Europe to buy Trade Goods and repeat until you have enough money to satisfy getting of to a good start. As with towns I like to expand at the rate where I can leave at least one piece of artillery in every town from the get go.
That is I don't build a new town until I have artillery and at least one colonist in "surplus" in a nearby village. Secondly I usually don't try to build many more colonies than Indian villages close by as these are usually important resources in the early game.
Later on, this changes more to the number of towns I can reasonably defend and/or towns close enough that trade between them is reasonably quick (i.e. not traversing large strands of map).
It really depends on your circumstances, if you get more people than a master farmer or fisher can support it's my rule of thumb to expand, but if it's too dangerous to expand, or it would spread you too thin, then don't bother, play it safe. In early FreeCol games it's better to play cautious then to go all out, because early slip ups will cost you your game, I mean, unless you don't mind restarting of course!
Try getting lots of money at the start by trading with Indian villages and then, when you have lots of money buy people and spread out so you have artillery in each village and that it is making hammers and tools. Then check your docks often to see if villages joined and take them by ship to your cities. By this method you will soon have a big empire which is big enough to destroy the royal troops and make you win. Although before declaring independence, make sure you have about 5 artillery in each city because sometimes the royal troops can break into your towns if you only have one. Although, at the beginning of the game don't worry about this and only get at least one artillery in each city.