What do I have to worry about when using a stainless steel pan as opposed to nonstick?

Food will easily stick if you're not careful.

That seems a little obvious and facetious, but it is a separate concern than cast iron vs. nonstick. With cast iron, a good seasoning reduces sticking; you just have to be careful to maintain it. There's no comparable treatment for a stainless pan (the closest thing is, well, a non-stick coating) and so you are almost inevitably going to get some adhesion, especially when cooking proteins.

This is not always a bad thing. A pan sauce, for example, requires some degree of sticking so that you can deglaze the pan and utilize the good stuff in the fond. Many proteins will also "release" once the outer layer is well-seared, and be easier to remove without sticking, assuming you can let them go that long without overcooking as a whole. You must generally take care to use enough fat, and you'll likely have trouble no matter what with delicate fish, eggs, and thinner cuts of meat (keep your non-stick pan around for those). I also recommend finding a decently thick pan with good heft, as a very thin bottom layer won't diffuse the heat of the burner as well and produce hotter/colder spots on the cooking surface; this can cause problems with inconsistent sticking and release.

Stainless can also develop a sort of "haze" on the bottom of the pan pretty quickly, which seems to be a mineral buildup of some kind. It doesn't really affect the performance of the pan, and it's easily cleaned off, but it can mar the shiny beauty of nice stainless.

Cooking: Sticking will be the most noticeable issue. It takes some getting used to, here are some starter advice:

1. If you are frying, make sure your meat/fish/veg is properly dry. A wet food will only worsen the sticking.
2. Once you put the food in the pan, let it brown for at least 2 minutes before trying to move/shake it. This will allow to develop a brown coat and will help release the food relatively easily.

Washing: Its more forgiving than a non-stick pan. A metal wool can take out most tough stains. Every once in a while, a cleaner like Barkeeper's Friend will help keep the shine.

If you haven't bought already, please take a look at try-ply clad or tri-ply base steel cookware, they are better at heat dissipation and usually of better quality.