If you have a linoleum floor, it may honestly just be better to rip it out. If any water or moisture got underneath there you would end up with bubbled up floors. However if that's not in your budget, or you're set on painting them, I would go to your local Ben Moore or Sherwin Williams and ask them. You may need a special type of primer to help the paint adhere correctly. -Kristen Vanderbeck at Design Spree

Do you mean real, decades-old linoleum or a more modern sheet vinyl with a clear wear layer on top? Photos would be helpful.

I successfully painted my very ugly kitchen floor - it was 70s era sheet vinyl. The steps are:
1. surface sand the ENTIRE surface. This is very important - otherwise the paint will not stick. And it's time consuming but you must do it.
2. prime with a good primer. It wouldn't hurt to do 2 coats here
3. Paint with whatever paint you like. For my project, first I made a layer of ceiling white because i had a lot of it. This was probably a good choice because ceiling paint tends to defend against moisture a little better. Then we made a spatter pattern on top to sort of mimic multi colored terrazo floor.
4. Polyurethane it. I used a super heavy duty (but quick drying) floor urethane - 3 coats - I hung out at the library between coats.
5. follow the directions on the polyurethane about traffic - avoid it if you can. Use ruby paper for the 1st week while it cures (not rugs they may fuse)

Note - be sure to completely clean the area and the adjoining area before polyurethaning. I was overly paranoid about fumes and I set up a virtual wind tunnel with fans, and I ended up with hairs getting sealed into the finish. But the spatter pattern is complicated enough that you cannot see them.

Also, the poly urethane I used did yellow some, but the warmth was actually a good thing in my case.

I have had this floor down for 8 years now. It was supposed to be a temporary quick fix, but it's held up beautifully and people are astonished when I tell them what I did. I've never had trouble with water (and I had a major kitchen drain failure that sent hot water all over with out damage.

Good luck with this. It will be great as long as you prep the surface.

PS: I also painted the ugly tiles - for that you need a kit from a specialized paint store. That was a nasty job, but has also worked perfectly

Rather than painting your floor, I’d suggest replacing it. Linoleum is very affordable and there’s lots you can do with it: Also you can use Vinyl composition tile and install right over your old floor for even less!