Dani, since new cabinets appear to be in order soon and this probably will not be one of those temporary projects that eventually turns into one with a 5 or 10 year lifetime, I would go as cheap and quickly as possible on it. Wash the cabinets with Dirtex, which is similar to TSP but easier on your hands, and rinse well with a sponge but don't leave pools of water standing. Sand with 150 grit sandpaper on a sanding block and just enough to knock down any shine and runs and drips, then rinse with a damp sponge again to get up ALL the anding dust. The next day, apply a coat of primer/sealer like one of the Zinnser products. As usual, I suggest working with a local paint store but one that carries either Sherwin-Williams or Benj. Moore paint. Let them tell you what quality to buy after you have told them all the details of your project and the same for the primer/sealer since that should be determined by the paint that you will be using. And don't forget to get and use drop cloths and painters tape where needed.
You probably will find it easier to roll the paint on with a roller rather than brush painting and it is faster. I would leave the doors on but remove the drawers since the old hinges are probably painted in place and removing them would cause a lot more work. Working with the drawers flat is just easier. With tinted primer/sealer, you may be able to get by with one coat of paint but that never seems to work for me.
Certainly, you could go to a lot more work and expense for things like stripping if you planned to leave the cabinets up for a while and that is your call. There have been many posts made here previously on that subject.