What equipment would I look for to use for starting out in time lapse photography?

I don't have a big budget to work with but I know the more money into a camera would lead to better images. I have just used my phone to take pictures but this has intrigued me. I'm just looking for something to learn with for now and I need to know if there is anything specific I should keep in my mind while shopping. My thoughts on what I want to time lapse are of the stars, moon, and distant storms(I found a great view point). Who knows WHAT I would think of when i get the stuff =)