Remove the trim around the edges of the flooring and pull up the carpet or vinyl. If possible, try to keep any trim intact so you can reuse it later on. If the trim is rotten, too, throw it out with the old flooring.
Assess the damage to the subfloor. Look for places that are rotten from water damage or excess pressure, or places that have holes in them. Make a drawing of the dimensions of the floor, making note of the locations where any pipes, cords or other features require special cuts in the floor.
Remove any rotten parts of the subfloor by pulling them up with a crowbar. Measure the thickness of the subfloor material (usually plywood) so that you can replace those portions with material of the same thickness.
Cut new subfloor boards to replace the ones you pulled out. Drill them into the framing under the subfloor, making sure your new boards are flush with any boards you left intact.
Cover the subfloor with self-stick vinyl tiles. Use a box cutter or razor blade to cut the tiles to size against the edges. Professional installers start in the center of the floor and work outward toward the walls --- or in the case of your travel trailer, the cabinets and other interior features.
Replace the trim around the edges of the floor, or install new trim if you discarded the old stuff.