Yup, you have to get down to a stable substrate before applying primer and paint. Adding new over top of flakes, will not make the flakes stay on. I hope that the rust not so severe that it's left holes in the metal - then there could be trouble if they're big and/or on the edges. If there are little holes, you might consider filling them with something like Bondo, then sand smooth before priming and painting.
Luckily there's no holes, just a lot of rust...
Do you know of a fast, easy way to get all of the paint off, or is just sanding it going to be my best bet?
Yey! No holes! I'd worry about using a wire brush on a drill or rotary tool, because I get a little intense and I might go to far. To suit my personality I'd go hand held wire brushes to get the big rust off, then a palm sander to smooth it out and for the paint. You don't have to get it all off, just get to where only stable paint (stuff that's stuck on well and not coming off as "dust") is left and feather the edges so it's a smooth transition between all the different areas before you paint. If you've got a good area to use a stripper like Naval Jelly, then that's a thing, too. I don't have that kind of space.
I think that's what I'm going to do! I tried to sand them the other day and gave up shortly after because there's a lot more rust than I thought there was. I've had them stored at my parent's since I live in an apartment so I haven't been seeing them every day haha.
Where can you even find Naval Jelly?