I would plan on priming and two coats of the new red. Have the paint store tint the primer to 50-60 percent of your chosen paint formula. As mentioned, use something like Benjamin Moore or Sherwin Williams and not box store paint. There deep tone bases are nicer than most.
You will want to prime with something other than white under your black trim if you go that direction too. I was going to try to talk you out of black trim but then remembered a really dramatic restaurant I did years ago in younger days. It had gorgeous lighting to pull it off though. Remember with such dark colors you will lose it rapidly.
True black paint does tend to cover pretty well because it comes factory mixed. Anything mixed at the paint store into a base is going to give you the same problems as mentioned with a dark red so expect to need two coats over the primer.
If I were you, I would play around with one of the paint companies free online virtual painting programs to try out your color combinations before committing to something like you have in mind as it will take some effort and paint to cover your choices if you do not like them. The Benjamin Moore one will even let you upload your own photos if you have the patience to mask them off (not complicated just takes some time). Otherwise, just pick a similar room from their selections. It will have lower and upper wall sections, ceilings, trims and all already masked off.
After that? You might want to invest in some sample bottles or quarts before going full bore. Paint out a few square feet and live with the colors in different lighting for a day or so to make sure?