I worked in motels for several years and wound up managing a small local chain. One of the things that was always a pain was the in depth cleaning that had to be done twice a year and one of the worst things to do was cleaning the grout and the caulking in the bathrooms!
The first motel I worked at had tile tub surround that extended to include the sink and did the grout and the caulking around both ever get nasty. Of course, we cleaned it the best we could between times, but when fall and spring arrived, we got after it with... ready? Toothbrushes and liquid dish soap.
That cleaned up the mess in a hurry with no awful fumes or dangerous chemicals. When I had my own home with a tiled tub surround, I made it a little easier and used a small scrub brush. It works, it's cheap and it's easy (well easier than some ways!).
If you're caulking in a bathroom, using the right caulk can make a real difference in how long it lasts as well as how hard or easy it is to keep clean.
Mildewed Bathroom Caulking explains the difference.