About a year ago, I prepared my concrete floors to be covered first with an overlay and then stained. My situation was a bit different, but I bet the prep needs to be about the same. You can read my blog post about the preparation we did on lauramakes.com.
For your situation, you will want to use a floor scraper to remove anything loose on the floor, like paint chips and the glue, or you won't get your paint or overlay to stick well. I found that wetting down the floor made the scraping go much faster.
You will definitely need to get the carpet glue up. I was interested in non-toxic products, so I looked into two soy based strippers. I tested Soy It, and it definitely worked for paint and is supposedly good for adhesives too.
After you get up the glue, you will need to take care of the grease stains. If you don't, it may show through your paint. The Eco Safety Products support person recommended that we use a floor buffer that you can rent to clean the floor first with orange degreaser and then water. Our grease stains didn't come completely out, but I don't think we found a heavy enough buffer to rent.
I can't really answer much about etching because I decided early on that the process was too toxic for us because we would be living in the house with our pets at the same time as we were prepping the floors.
Anyhow, I hope that helps a bit and let me know if I can elaborate on anything.
You could take a proactive approach and sand it in any case. This will lead to better paint adhesion which is a good thing. It is also a great idea to wash previously painted surfaces with a strong solution of TSP (tri sodium phosphate). Make sure to use good rubber gloves and hot water - for your protection and best results. The TSP wash does a fantastic jub of getting any gunk off the walls that can interfere with a good paint job. Especially in a bathroom there is always the possibility with soap film and over spray from foam cleaners on the walls and panelling. A strong TSP wash can also remove some of the gloss from a previously painted surface for some types of paint.
Best Answer
Use one coat of a clear or white latex primer. I'm assuming you meant an opaque paint with an eggshell finish, not a stain.