Once you get the old tile up, sweep/vacuum until subflooring is completely clean. Self-stick vinyl tiles are probably your cheapest, easiest option. I especially like the new simulated wood flooring. And the vinyl has come a long way, very durable and pleasing to the eye.
Use a spud! It's a heavy push tool that will lift up the old tile as you push it along the floor. Depending on the method used to adhere it in the first place it might be surprisingly simple... or not! If the tiles come out in one pice you can offer them to some place like Habitat for Humanity, recycle or you can just dump them. Home Depot and Amazon sell spuds or you might even be able to rent one. We removed 900 sq yds of asbestos floor tile using one! The entire lower level of our home was one big asbestos mess!