Piston-valve brass instruments can be cleaned easily with lukewarm water, soap, and a bit of elbow grease (scrub brushes for leadpipes and valves)
Be very, very careful in your choice of soap. You want one that has no scents, no extra chemicals (like OxyClean, bleach, or "rinse aids") or anything else. Pure base level blue Dawn is good, but if you want to be sure, buy the regular bottom-shelf contains-nothing-but-cheap-soap Ajax.
Anything with a rotary valve is entirely up to you. If you're comfortable re-packing valves, go for it. If you're not, leave it for a pro.
And woodwinds? Leave clarinets alone unless you know what you're doing. Same with saxes - the rods and assembly can be a beast. Never touch a double reed unless you're a pro.
But a flute? If you strip off all the rods and keys, you can actually clean a flute pretty effectively. Hope you know how to put it back together and level it, though!