Some of the less expensive finishes on newer DBs (like mine) are nitrocellulose lacquer. I've found that Gibson's pump action guitar polish formula is fantastic for preserving this type of finish. It actually has a touch of lacquer in the polish so it reduces the very light scratches and finish cracks and removes the fingerprints. It also makes the finish somewhat rosin resistant. Like Ken said, it's not something you want to let soak into a crack in the wood (to be distinguished from the light crazing that eventually appears in aging nitrocellulose finishes).
For getting the heavy gunk off the FB, I use Formby's furniture cleaner on a piece of new rough cotton canvas, but I am careful not to get that on my strings. I don't know that it would hurt them, but it is made for cleaning wood so I keep it on the wood. After that I put some lemon oil (Also Formby's) on the FB. Definitely keep any oil away from the strings where you bow them or you will have a very hard time getting the friction back. Any organic (alcohols, etc.) solvent is going to evaporate some of the oil in the wood, so like uncle Paul says, put that back with lemon oil.
Powerchord "Skank" Remover (hey, that's what is says on the bottle) gets that stuff off the neck handle without strong solvents.
There is a very good metal string cleaner made by Dunlop that you can get from most guitar shops. Of course if you use gut strings, you need to use something different and there are various discussions about that under "Strings".
Petz makes a rosin remover that works almost as well as their rosin.
If there's little glop pimples all over your bass from using soft sticky rosin, you can safely get it off with the rosin remover. Use a dry hard cloth to wipe rosin powder off the strings.
Greasing your tuning machines (you probably only need to do this once in ten years) will make them work smoother and last longer. Bicycle bearing grease is what I used.
I know this thread is primarily about cleaning, but proper care should also include a padded bag, a sturdy bass stand, and a room humidifier for the dry days. Also I've found it a good idea to get about an acre between me and the drummer's hardware.
There was a great article on David Gage's site that explained all the safe ways to move and transport a DB, but I couldn't find the link. Do not use the f-holes as "handles". The best lifting points are at the C bouts and the neck heel. Avoid sliding the bass even on carpet, as you can rope burn the varnish off pretty quickly. Also resist the temptation to let people "help you with that" when you are carrying the bass. I've found that a second person in the operation usually just throws me off balance and makes it trickier to carry. Also those not used to moving it don't know to watch the scroll around the low doorways. Also avoid rehearsing in basement studios that have low hanging duct work and plumbing.
You are an encyclepedia of bass, and good thing.
