How soon after baking do you remove the cake from the pan? Lately it seems that my cake (mostly white) is getting soggy on the bottom after it cools.

You can take them out about 10 minutes after they come out of the oven, just give them a little time to cool a bit.

It depends on the type of cake, but most cakes I leave in for 10 minutes, and no more. I do have a few recipes that require they be flipped out immediately. The only type of cake I leave in longer is angel food, because it's required.