What's the best way to remove old vinyl tiles and adhesive from a basement cement floor?

Elbow grease! You can pop the tiles with a heavy chipper like and ice chipper or flat shovel. The adhesive is another matter. How clean to do need to get the floor. Jasco adhesive remover and a 4" razor scarper will do the job but it's a ton of work. Check around at rental store to see if you can get an electric floor scarper for your job. Homax Products - Cleaners/Removers - Jasco Sealer & Adhesive Remover