1) All valves should move smoothly and quickly. They should also not have any dents, major discoloration, deep scratches, or flaking metal on the valve itself.
2) All slides should move freely. First and third slides should be smooth and fast while the second and tuning slides should be smooth but slow moving. The slides themselves should not be extremely discolored or have any dents or major scratches. No properly oiled/greased slide should have a “scratchy” sound when moved.
3) The inner tubing should be free of mold or major discoloration.
4) There should be no cracks through metal or major dents in the body of the instrument.
5) There should be no sign of any kind of corrosion through the metal. Specifically look for Red Rot which is identifiable as a red spot on the instrument that is circular-ish in shape, with a dark spot in the center. However, wear in the coating of the instrument (through the lacquer, silver, or gold plate) is expected and harmless. If you see brass it’s ok. If you see red with a black dot its a problem. The attached image is a classic example of Red Rot.
If you are unsure about an instrument ask your teacher. However, in general, if something seems off it probably is.