A tuba is quite large, and can be quite a challenge to clean thoroughly on both inside and outside. I need some good advice on how to do this properly without harming the mechanisms.
This is for a piston-valve tuba, by the way, not a rotary valve. Also keep in mind to do this all very gently, tubas scratch easily.
You'll want to take it completely apart (valves out too, and remember how it goes together) and set in a bath tub full of warm/hot water. Then, add some soap (mild, non-abrasive) and rub it gently with a cloth to get off grime (If it was particularly dirty inside, you could try a toilet scrubber(new) and gently clean out the bell (or use a cloth preferably if your arm is small enough), and use the snake to clean out the finer tubing. Make sure to clean the snake often so you just don't spreak the grime). Once it has been scrubbed, drain the tub and then rotate it head-over-heel a few times to get all of the water out and set it on a bath towel to dry. While that is drying, place the slides in the soapy water and rub them gently with a paper towel / cloth, then dry them off and set them somewhere safe. To clean the valves,take off the felt and rubber bumpers and wash them in SOAP-FREE water, and don't use the snake to clean out the holes in the valves. When all parts are COMPLETELY dry, you can begin assembling your tuba. Reassemble the valves(IN THE SAME ORDER THEY CAME OUT), making sure to put on plenty of valve oil, and grease up the slides before you put them in also. (put on the grease with a paper towel, you're not supposed to contact the slide because it can deteriorate the metal)
Another tip, the valves should be cleaned about every month or two, but missing a cleaning here or there won't hurt too bad.
For a rotary-valve tuba, the process is slightly different because the rotary valves cannot be removed from their housings (unlike piston valves). However, the basic idea is the same:
More often than bathing your tuba, you should boil the mouthpiece. This is more like a biweekly thing if you play a lot, especially if you don't have the chance to brush your teeth or at least rinse your mouth before beginning to play. This process is simple; bring a pot of water deep enough to fully submerge the mouthpiece up to a boil, and using a pair of tongs or some similar apparatus, lower the mouthpiece into the boiling water. Boil for about 5 minutes, to loosen the gunk and grime, then remove it and run it under lukewarm water until the mouthpiece is cool enough to handle. Then, run through the throat and backbore with a mouthpiece brush. Rinse, then repeat if desired, as necessary to clean out all the gunk, before letting the mouthpiece air-dry completely.