Here's info to help you make your decision:
PS3 cannot play PSP download games. Neither can PS4.
PS Vita cannot connect to a TV
There is a separate device coming out in the next year or so called PS Vita TV which is a non-portable Vita that outputs to a TV and uses either a Dualshock 3 or Dualshock 4 as the controls. Should retail around $100.
PSP's TV output does not take up the whole screen. It's got black bars all the way around. Like so:

There are Chinese-produced unofficial adapter boxes which will upscale the PSP video output to take up your full TV screen. I don't know a lot about them other than that they exist.