I just bought a lil Morgan, one very level headed lil guy for being 4. Completely the opposite of the age/breed I was looking for but he's perfect for me. My trainer picked him out based on his training/personality and how it would mesh with mine. Took her 6mo to find a suitable horse, we literally considered hundreds.
Find a trainer you like and take some lessons. Then when you are ready, have the trainer assist you in find the perfect horse for *you* and don't buy anything the trainer has a financial stake in, if the trainer is pushing you to purchase from them, find a new trainer!