There were two layers of vinyl tile on the floor in the back half of the basement: one self-stick (visible in the garbage bag below), one old VCT (same stuff we had installed in our kitchen, but much less beautiful, shown in the second photo).
Because we did not know the age of this flooring, we knew we needed to remove it with care because of the possibility of asbestos. After some research, we determined that if we kept the tiles intact, did not sand, and worked in a damp environment with ventilation, we would be okay.
So, we filled our utility sink with hot water, soaked some old towels in it, and laid those out on the floor as we worked with putty knives and a flooring shovel to lift the tiles. The heat and moisture from the wet towels was very, very effective at loosening the vinyl adhesive (which perhaps had loosened from age and wear as well). Less than five hours later, we had a bare floor and 6 garbage bags full of trash.
The next day, I thoroughly vacuumed, wet mopped, and vacuumed the bare concrete. Then I painted. (Behr Porch & Floor Paint in silver gray, low sheen finish.)
Here you can see the results of two coats of floor paint (minus a few spots where big furniture was in the way -- those will get touched up later), plus scraping the paint of the windows.
I painted the stairwell between the basement and kitchen as well. I used leftover yellow paint from our kitchen bead board.
Now for some electrical work...the professionals are scheduled to be here in a couple of days. The end is drawing nigh!