If you suspect an aklyd (oil) paint, you should defintitely do a "scuff sand" of the entire surface, not really a sanding so much as a scuffing - a kitchen scrub pad such as a Scotchbrite will suffice. The idea is to roughen the surface so the new paint has some roughness to physically engage onto.
Don't paint alkyd over emulsion (latex) unless it is very old, and you do lots of surface prep.
If you are 100% sure it is alkyd, you do have the option to stay with alkyd. This gives you a very tough paint that will stay easy to clean. damage that irrevocably mars a satin latex will wipe right off an alkyd gloss. However it will tend to be Very, Very Smelly during the first few days after application, and the family might have a big problem with that.
Check your laws, for instance in the United States, all architectural paint must have extremely low VOC (like 200g/l, impossible for any but emulsion "latex" paints) -- unless they are sold in quart or litre cans, an intentional loophole to allow folks who are "serious" to do what they want. Freedomz(tm)! And that exception does not exist in the South Coast Air Quality Management District of southern California.
You don't want to strip a concrete surface down to bare. You really don't. Or a drywall surface.
A primer will give you uniformity of texture. Without a primer, places with a different surface, e.g. Spackle, will soak in differently and present a different surface with different texture and gloss. Primer makes the entire surface the same. It also helps hide the old color, though, it is not for that.