Andy, how big are the tiles? IS the glue used to hold them down black? In the old days they used asbestos to make the tiles. If they are smaller then 12"x12" they are asbestos. If they are 12x12 and the glue is black they are probably asbestos. If they are 12x12 and not black glue, they might be asbestos. How big is the area you want to do?
They are fine to work with and remove. But you don't wand to sand/grind on them, making dust out of them would be bad.
I would recommend ceramic tile. Easy to do yourself, durable as all get out, and fairly inexpensive. All you would need to do it take up the rest of the tiles, scrape up the glue as best you can, and set the tile.
Home depot (and others) sells nice 12x12 ceramic tile for as little as 89 cents a foot. For 2 bucks a foot you have many to choose from. If you want more tile info etc shoot me an email.
Another choice would be a floating floor. All you would have to do is fill in where the tiles where missing and install it. Price starts around 3$ a foot and goes up (i think, i don't do this regularly so the price quote is a guess). The problem is that it has to be able to move, so you need a 1/4 inch space all the way around. You have to cut off your doorframes so it can slip under, you have to remove the base boards and then reinstall them to cover the space (or add 1/4 round moldings). Also this type of floor isn't as tough as tile and always feels/sounds hallow to me.