Does the type of paint previously used influence what I can used next?
Yes, it can. Here are some things that can come into play:
1) How well adhered the current paint
is (important no matter what you are putting on!).
2) Whether the current paint
is a harder (usually two-part) paint or a softer paint.
3) Whether you are looking to put on a harder paint (the general rule is that you cannot go harder over softer).
4) Once you have a certain brand/type of paint chosen, they usually have tests you can do to determine if the current paint will be compatible as a substrate. For example, a typical test would be to crosshatch the current paint in a certain way, and then tape a cloth soaked in a particular thinner over that area for "x" number of hours.
5) If you are interested in using a softer, one-part type paint, then you don't generally have to worry about #3 and #4 Again, once you choose a paint, there will be specific instructions, including what grit paper to sand with, etc. (of course it's a bit of a vicious cycle until you've chosen a paint, I realize).