I normally use rubbing alcohol most times, but I'm not a fan of heavy stains. But if I'm trying to keep the stain I would use wet toothbrush or q tip (great for rims), then wipe with a fluffy rag. I know folks say briar and water don't mix and prolonged water contact can make the grain swell, but a little bit doesn't hurt it in my opinion.
You might also take a look at some stiff bristle shoe brushes. After that I do like Sam and use white diamond (unless it's got heavy rustication). You'll be surprised at the shine you can get from the polishing compound, cheap wheel, and a cordless drill.
I *just* started trying Carnuba wax on my last two restores. Not sure how I feel about it yet.
One tip I never see mentioned. Once you finish the clean up make sure you give the inside of the bowl a wipe down with alcohol. Lots of junk from the cleaning (briar dust, compound, wax) can get inside and give you a funky tasting smoke. I also never like how they smoke the first time or two. I can still taste the alcohol usually.
I'm still learning but there are some awesome refinishers here on this site. Show us some pics!