To remove old resilient flooring, first cut it into parallel strips about 6 inches wide with a utility knife. Use a hammer to tap a stiff putty knife or brick chisel under the linoleum to break it loose. Pull the linoleum up in strips to reveal the backing or the glue. Once the surface layer is gone, use a paint scraper to remove the glue. You can also use a heat gun to soften the glue as you scrape it away with the paint scraper.
Some old linoleum has tar-based adhesive, which can be softened by applying mineral spirits.
To remove the linoleum and subfloor together, drill a hole through the floor to determine how thick the plywood is. To cut flush against the walls, use a reciprocal saw, but be careful you don't cut the floor joists. Cut the floor into manageable sections about 3 or 4 feet long as you continue to remove it. When laying down the new subfloor, nail crosspieces between the joists to support adjacent plywood subfloor edges, particularly if the old floor was tongue and groove plywood. Peel away enough covering in a corner until you can judge which way the flooring runs.