Most Tropico 2 campaign missions have similar objectives. Usually accumulating a certain amount of gold, in your personal stash and/or for the island. Therefore a general strategy is applicable to most missions.
A new island, in campagin levels, will generally not have a Timber Camp, so it is important to not forget to place one of these, near dense trees and close to the Sawmill. The 20 lumber that you start off with should be used to build a second Sawmill, either next to the initial one, or in another area with dense forest. Place a second Timber Camp next to this. Producing lumber at a fast rate is a primary aim at the start of all levels, as lumber is usually the limiting factor which slows down the development of your island and ship building. Each Timber Camp can supply two Sawmills. In only a one-to-one ratio, the Timber Camp will progressively build up large amounts of wood, which one Sawmill cannot process quickly enough. So building at least a third sawmill early on is recommended.
The second main aim at the start of any level is to send a ship out repeatedly for skilled and unskilled captives. You should be able to send the starting ship out roughly ten times before pirate happiness drops below 50%, in which case the pirates should be allowed to fulfill their entertainment needs. It's good to accumulate a large number of captives, even more than the stockade can take, even if you dont presently need them, as they will be required as you build more on the island. Furthermore, later in the game you want to focus primarily on cruising. The levels where your Patron sends captives on a regular basis are far easier, and the initial use of the ship can be for capturing skilled captives.