Forsest TIPS:
1.) Leaving Forests: Unless your expansive, you will want to save at least 2 forests per city. If your not expansive and you have an unconnected inland city with no forests, it has 1 health only... pretty bad.. ALso, not so later on, lumber camps give +1 hammers which is pretty good.
2.) Overlapping: People say its bad. I duno why. How many cities have you gotten to 21 population to use all the sqaues. Overlap your cities so that they share forests. If 2 of the same forest sqarues are in the limits of both cities, both cities share the health bonus. You could chop generally in the close square of the city and leave the outside squares to share with other cities.
3.) Cut all forests near rivers: Everything is better about this. Any mprovement other than forest yields +1 commerce near a river. Also, if you build a farm, yea its the same power (3) but it becomes 4 after biology. Also, if you build a mine or a watermil or a workhouse, they get upgraded with tons of bonuses. Lumber camps and forests do NOt get the +1 hammer bonus when a railroad is on it.
4.)Plain inland forests: These are the ones to keep. In thte early game, chopping reduces the square power from 3 to 2 with the same amount of measly food. After lumber camps they become like a grassland hill which isn't to bad.
5.) What to cut: You shuold really only chop non growing units like workers and settlers but wonders are ok if u need to get it fast. If you chop warriors or anything else, your wasting hamers and while building these units your city could be growing. If your city produces 8 or more hammers, its pointless to chop for the city. It only takes 4 or 5 turns from the process and you should save the forests instead.