You can get a free colonist for only 600 gold from Europe. You're also going to get a lot of them from food and from Indian converts after Casas. You need to educate them to get decent performance from them.
Note what skill is being taught in every Indian village you visit, even if it's relatively far away (you can use ships, but unfortunately not horses, to get there). If it's something useful, send a colonist (or even better, and indentured servant) there. Even if the skill is useless, you can send a servant there and then clear their profession to get a much more valuable free colonist.
Remember that you need to get some skills (Master Sugar/Cotton/Tobacco Planter) from Indians before you can have your colonists teach the others - those experts can't come from the Old World. Sometimes a Free Colonist doing one of these jobs for a long time can become an expert spontaneously, but it's rare so don't count on that.
Each Indian village except for capitals can educate only one colonist. Capitals can educate as many as you want.
Some people reported that Indian villages sometimes change their specialty or teach a different skill than they report. I haven't observed a single instance of this, but I haven't tried too hard. Maybe it's a bug in some versions.
Build a schoolhouse as soon as possible. You can train extremely important Farmers, Fishermen, Carpenters and Pioneers there. You probably won't need many scouts, so it's usually easier to get them by immigration (and/or by learning from Indians).
Ore Miner and Lumberjack are easier to get from the Royal University in the late game, but it may be cheaper to train them while the immigrants are still inexpensive.
A College is also very useful - it doubles your teaching capacity and lets you train Master Blacksmiths and Master Gunsmiths. It will probably be easier to make your soldiers veterans by fighting Indians (after getting Washington), but you can educate them too.
The only good skill taught at the university is the Elder Statesmen. Forget about the other two (Firebrand Preacher and Jesuit Missionary). The other useful thing is that you can teach 3 students at the same time.
Professions other than those specified above are less crucial, but of course it's better to have specialists work on other things too.
The time required for teaching is independent of revolutionary support and buildings in the city, but depends on the teacher:
- 4 turns for Schoolhouse-level teachers
- 6 turns for College-level teachers
- 8 turns for University-level teachers
It depends on who's teaching, so an Elder Statesmen teaching a Criminal/Servant will take 8 turns, while a Farmer teaching a Criminal/Servant will only take 4.
You can "accumulate" the teaching. The student only has to be in the colony when the counter hits whatever it's supposed to.
Think of it as 3/5/7 turns of "doing research", and 1 turn of actual teaching. If there's nobody to teach when the counter hits, the research is lost. Every teacher has his own counter, so 2 Statesmen will teach 2 colonists in 8 turns, not 1 colonist in 4 (you can move second Statesman in when the first's counter is on 4 if you want this behavior). I'm not sure how moving teachers in/out affects their counters, or how to get the teacher with a lower counter out of school if you have 2 of the same kind that started in different terms. To do that, you need to pay attention and note the position of teachers on the roster tab, not in the university itself, and change their occupation by clicking on them there.