The best you can do by hand is to rub it with a small folded-up piece of cloth and Brasso. For very encrusted spots, you can put some Brasso on fine steel wool and clean them with that. Then you'll have to restore the polish with the cloth and Brasso. All this is assuming that you don't have access to tools like a vise, electric drill, and buffing wheel. If you can get a buffing wheel and put it in your electric drill, you can polish the areas with Tripoli, a polishing compound available at Sears and hardware stores.
Tripoli is a coarse polishing compound made of pumice embedded in a wax cake.
After this coarse polishing is done, you need to clean the surfaces you've polished, and then polish again with another buffing wheel with jeweler's rouge, a finer polish made of iron oxide. You can get a cake of this at the same place. It's better, if possible, to clamp the drill or somehow fasten it securely and hold the trombone against the wheel, rather than trying to hold the drill and control it. After you've polished the trombone, it has to be absolutely clean and bone dry before you spray it or the lacquer won't stick. If you clean it with water and detergent, it will tarnish the brass.