I don't need the money (I just don't like getting shortchanged and screwed).

Aren't you delightful. You could have negotiated or not taken the spot in the beginning. It's also not as simple as getting shortchanged/screwed. You realize an internship is a parasitic relationship, right? You literally are there to take money from the company while learning. You are not going to contribute to them in a positive way. You're charity work.

So yes, pay sucks. But it's because you're there learning in the hopes you come back in the future for a job when you'd then help them make money. It's also a startup where they can't afford to pay interns money.

All around dick move and all around dick attitude.

So tomorrow I'm giving my 2 weeks' notice.

Honestly you haven't been there long enough for 2 weeks notice in addition to being an intern. You should ask them how long they want you to stay on. Probably don't need you past a day.

How do I avoid telling them anything about the position I'm starting

You can literally just say you are leaving for personal reasons. That you can't say and appreciate the opportunity they gave you. You aren't legally forced to say.