should I get just a MIDI controller or a digital piano.
A MIDI controller won't have a built-in synthesizer, so you would need to have it always connected to the computer, and you'd need some software synthesizer with good piano sounds.
It might not be easy to configure the computer for latency that is low enough to feel natural.
I'm looking for the most realistic piano sound and feel(weighted action I guess?).61 keys minimum.
The sound depends not so much on the price but on the age of the piano; recent devices have better synthesizers and more sample memory.
Weighted doesn't mean much; realistic feel requires actual simulation of the hammer action. Look for the word "hammer".
I don't know of any device where you get this with less than 88 keys. (And there goes your budget ...)
You might get either a digital piano, which has its own stand, or a stage piano, which is designed for mobility, so you'd need a separate stand and probably pedals.
Have a look at devices like the Casio CDP-100, or the Classic Cantabile SP-4 or DP-40. (I'm listing these because they are the cheapest that I could find. You might be able to find cheaper no-name pianos.)
In any case, I'd recommend to check them beforehand for their keys' feel, if possible.
connecting with PC won't be bad either
All modern pianos have MIDI or, more commonly, USB.
