Sid Meier's Colonization/Immigration

You want to get as many colonists as possible, as fast as possible. There are a few ways of getting them.

You get immigrants if your churches make enough crosses, or if you pay for those crosses. It's the main way of getting colonists at the beginning, but they are often of low quality. Indentured servants should be sent to Indians for education, but petty criminals may be more of a burden than a value. Get William Brewster in Congress as soon as possible - no more servants and criminals, and you get to select one of 3 available immigrants.

If your scout finds a Fountain of Youth, you get 8 immigrant colonists for free. It's not unusual for this to happen 2 or even 3 times during a game, if you explore a lot and have some luck. The chances don't seem to depend on difficulty.

As the game passes, each immigrant costs more and more. Because of that, major church-building programs are a waste of time (and the English are not as great as they seem). If you get a Firebrand Preacher anyway, building a Church for him is reasonable. But better think about the economy. I haven't done any calculations on the crosses, but I've seen some suggesting that each colonist in a European port reduces number of crosses you get each turn. That's one more reason for getting them out of Europe as soon as possible.

You can get a free colonist for 200 food in any of your colonies. It's a slow way, but you're likely to have big food surplus, so why not do something useful with that food. It's usually more practical to move all the food into central cities, far from Indians, so it's converted to colonists faster, and the Indians won't steal it from you (and they can even give some food to you).

Paying for educated workers at the Royal University can be helpful.
 * Get an Expert Ore Miner and clear his specialization. Free colonist for only 600 gold. That's the way of getting colonists in the later stages of the game. Then you train him in whatever profession you want.
 * Ore Miners, Lumberjacks and Gunsmiths are so cheap, getting them there is more efficient than training them in the colonies. Silver miners too, mostly because you only need a few.
 * The rest, except for the last four, are in a reasonable 950-1300 price range. It may be cheaper to get them trained instead of training them yourself, especially if you need a lot of them quickly.
 * You need at least one colonist with a given profession to teach others. The exception is that cash-crop farmers can only be taught by the Indians; everyone else you can get from the Royal University.
 * Veteran Soldiers are unbelievably expensive. Get a free colonist for max 600 gold, arm him (arms are cheap early, you'll be making a lot in the colonies later), give him horses (you'd have to even if he was from the Royal University), and attack some Indians. With George Washington, he will be promoted if he wins (and only lose 50 cheap horses if he doesn't). That way you can get almost 3 veteran soldiers for the price of one from the University.
 * Elder Statesmen are very expensive, but sometimes necessary. Teach your statesmen if you can (unfortunately a University is required).
 * Jesuit Missionaries are a waste of time. With Jean de Brebeuf all missionaries function as experts, even the petty criminals.
 * Firebrand Preachers aren't very useful either. Immigration gets more expensive with each immigrant, so their value diminishes quickly.

You can get a few colonists by conquest or by Indian conversions. Converts aren't very useful, except in the early game. Bartolome de las Casas turns all your converts into free colonists, but that's just a one-time action.

Other powers develop so slowly that you wouldn't get enough colonists from them even if you conquered all their colonies. But small attacks here and there are a good extra source of colonists.

If you combine all strategies - a few churches, paying for immigration or at the Royal University, search for fountains, Indian converts, food conversion, and conquest - you will get flooded by colonists, and with proper education and organization you can expand at an astonishing rate.

Prices at the Royal University
Generally Schoolhouse skills are cheapest, and University skills are most expensive, but there are some exceptions (expensive Farmers and Soldiers, cheap Gunsmiths). Exact calculations in the next section.

Veteran Soldiers get 50 free muskets (at price 6 it means 1700 for the Soldier itself) and Pioneers get 100 free tools (at price 4 it means 800 for Pioneer itself), so their actual price is lower than their listed price.

Value of education
Because we know the price of a Free Colonist (600), prices of already educated colonists, and the time needed to teach them (4, 6 or 8 turns, depending on who's teaching), we may calculate how much the teacher's time is worth.

The value of Muskets (6*50=300) and Tools (4*100=400) at some arbitrary price level has been subtracted from the Pioneer's and Soldier's prices.

In the early game, when colonists are cheaper than 600 gold, teaching is worth even *more*.

Value generated per turn:

For comparison (on good but not ideal terrain):

As you can see, teaching can be extremely profitable (or to put it other way, paying for training extremely unprofitable).

In particular, start training your own Expert Farmers, Expert Fishermen and Master Carpenters as soon as you can (Schoolhouse is cheap, and few other professions create a value of 100 or more per turn), then Master Blacksmiths in College, and Elder Statesmen as soon as you build a University.

Expert Ore Miner, Expert Lumberjacks, Master Gunsmiths and Hardy Pioneers can be bought - their education is not that valuable.

Veteran Soldiers seem like their education is worth a lot, but that's not really true, as you have a third source of them - Washington and attacking Indians.

Immigrants to select first
Seasoned Scouts and Hardy Pioneer are probably the most useful. Expert Farmers and Fishermen are also very important to have, especially since one such colonist can teach others.

You need specialists for your cash crop. The only feasible way of getting them at that point is taking either a free colonist or an indentured servant and sending them to the right Indian village.

If you're very lucky, a free colonist set to growing a cash crop might become an expert, but you cannot count on that happening. Still, until you can get such a specialist, you might be best off having a free colonist (rather than specialists or criminals/servants) grow them, on the off chance that the game promotes them.

Get Brewster as soon as possible to improve the quality of your immigrants.