My anecdotal, not necessarily rational evidence:

I used store-bought wart remover, used it once in the morning for a few weeks until it looked like it had gone away. It recurred. I tried the store-bought cryotreatment, but it didn't seem to have much of an effect. I got more regular wart remover and used it again, this time using it about two weeks longer than it seemed necessary. This worked, with virtually no trace now. But I'm sure every wart is different.

With the normal wart remover, every week or so, you should go ahead and try to peel away the dead layers. Be careful not to peel too far away from the wart itself, but this will happen. When it does and you have raw skin around it, use some sort of ointment on the skin around it to prevent the acid from eating away more of it. I believe I also took some days off and just used antibiotic ointment on it. Covering it with a bandaid is helpful until it's pretty far down. The remover gel dries and smooths so you don't really need a bandaid for the last few weeks, if you take care not to get it too wet.