I kinda like the counter tops -- retro.

Are you sure the tile is plastic? If so, I'd go to a local paint store that painting pros use (i.e. not a big box store) and ask them what they recommend. But if this is a heavily used bathroom, you might want to think twice about painting the tile as I doubt it would hold up over time. Maybe some pros can advise on products.

In my current house the bathroom walls were in bad shape from years of poor quality "improvements" and repairs. Instead of having new sheetrock installed, I used bead board. Looks great in an older home and it's affordable.

In another home I owned I had a vanity similar to yours and I painted it, but I skimped on the prep work and it wound up with chips and scratches within a year. If you decide to paint it be sure to scrape and sand loose paint, lightly sand the entire cabinet, clean it thoroughly, and prime it. Then paint it. Two coats. You'll probably need to scrape off old paint on and around the doors to get them to close properly.

If it were my house, I'd probably opt to paint the exposed walls and the cabinet, buy some new towels, then save my pennies for a more extensive remodel.