Hey!
I've been playing for about four years now, 100% self-taught.
The first piano I bought was about £99 from Casio. It was a 61 key bottom-of-the-range one, because pianos can be expensive and it's a lot of money to spend if you don't end up getting along with it! The most important feature to look for in my opinion is to make sure it has a headphone jack (mine came with headphones and a stand) so you can play without disturbing anyone or feeling self-conscious.
I, like you, was conscious about how much money I would be spending, and although it was a little bit harder to get to grips with, I learnt a lot of stuff from YouTube. Channels like PGNPiano or PianoKeyz use really helpful displays in their videos and I just picked easy versions of songs I liked. Once you learn to read sheet music, YourPianoCover is a really cool YouTuber as well. However, I imagine your life would be a lot easier if you went onto Amazon and searched for some Grade 1 Piano books, as this will explain the theory that a tutor would.
I would start out learning it as a bit of fun - a party trick almost. Ignore sheet music and key signatures and stuff like that until you're comfortable with playing a few songs. When I learn a song, I like to play around with it for a while, try playing some bits faster or slower, changing the 'voice' option and trying to make the song fit into different situations, just anything that will get you more comfortable to playing a piano.
Hope this helps!