The AI is extremely dumb. The same thing can be said about the AI in all the classic strategies except Freeciv. You can easily win any war against other powers even with a greatly inferior military force. Don't even bother keeping armies to defend your colonies; Indians are a much greater problem than all the 3 other forces combined.
Another great thing about the AI is that it very eagerly signs peace treaties with you, and very rarely breaks them (but remember that attacking a colony where Privateers stay does not break a peace treaty). So one bad-ass strategy is to find a small enemy colony, extort money from them, attack the colony anyway, and sign the peace treaty after you capture it.
The AI cheats. At least one confirmed cheat is that it gets 20 free tools and 50 guns in every colony it starts (but I'm not sure if that's immediately or after a couple of turns, and I'm not sure how much it depends on level).
As a proof of AI stupidity, I've seen AI deforesting 2 Prime Timber squares next to a single colony. It is also known to plow all forest squares surrounding its colonies, and bafflingly insists on founding colonies right next to Indian villages.
This is a very useful way of getting valuable goods, tools, and weaponry. One way to simultaneously defend yourself and weaken your opponents is getting one or two Privateers to loot their merchant fleets.
With Drake they get a +50% strength bonus, and a ship loaded with cargo (your target) gets a penalty, so you aren't likely to have much trouble.
You can attack even if you're not at war. If one of the other powers demands that you return all your privateers to Europe, you may as well do that. You can then immediately order them to return and they'll be back in a few turns.
The Privateer is also a decent transport ship (it's small, but very fast) and an excellent vessel of exploration. Investment in privateers will pay off handsomely.
Unless you need to get rid of a boycotted good, don't bother trading with other powers. It's better to take what you want and assimilate their colonists. In Colonization it's the Borg way that wins.
Remember to sign a peace treaty when you don't need a war (it's always an option after Franklin). You don't want your colonists attacked by reinforcements sent by their king. They aren't going to significantly harm you, but may be annoying and it's a waste of resources to keep a standing army everywhere with your colonists just because someone may get the idea of striking there.
One word of warning - you can attack any city with a Privateer inside without breaking a peace treaty. So don't dock your Privateer in a city with no defense and enemy Dragoons nearby.
You can probably do the same thing, but why bother if you can sign a peace treaty again afterwards?
I don't think this is documented anywhere, as even one of the Colonization FAQs available online seems to be unaware of this and calls it a "Colonization quirk". Just don't worry :-)
In mid game, others' colonies will become hard to conquer.
1 Artillery in a Fort (+150%) has strength 12.5. Even something as weak as a Non-Veteran Soldier in that Fort has strength 5.
So if you attack a Fort with 1 Artillery and 4 Soldiers (or more likely 2 Soldiers and 100 stockpiled guns) your expected loss is the damaging of 3.8 Artillery or horse loss by 6.6 Veteran Dragoons.
If they have 2 Artillery and 5 Veteran Dragoons the expected loss is 12.1 damaged Artillery or horse loss by 21.2 Veteran Dragoons (that is - expect to lose 1062 horses).
It's very unlikely to be worth the investment.
You may think that fortifying Dragoons around a city to starve it may be a good idea (the squares on which you fortified can't be used), but 1 Expert Fishermen can provide food for 4-7 citizens and 2 are fed from the central square, so if the colony had 3 ocean squares you will fail, and in case of smaller colonies just a single such square is enough.
So don't attack bigger colonies unless you really have to. Privateering (with Privateers or even Frigates), attacking small colonies and troops/colonists in the field, money extortion and maybe even trade are usually worth much more.