I inherited a bunch of raspberry plants along the fence in my backyard.
How do I care for them? And prune them? Will they come back year after year?
They will not only come back year after year, but most likely they will spread vigorously if you don't keep them in check.
As for pruning, it greatly depends on whether your raspberry is primocane-fruiting or floricane-fruiting. I've linked to the Cornell extension's fact sheet, which will tell you what to do in either case.
Here is a link that might be useful: Raspberry Pruning (Cornell Extension)
Raspberry canes live for 2 years, and for the most part, they only produce fruit the 2nd year. So at any given time, you will have 1st year canes just beginning the cycle, 2nd year canes that produce fruit, and older canes that are dead. The old dead wood should be removed immediately. The 2nd year canes should be thinned to 5-6 canes per foot, keeping the strongest ones that are within the row that you want to maintain. Dozens of others will probably need to be removed. But keep all the baby canes that start out sort of purple in spring then turn a brighter green. If they spread too far from your row you can prune them out but let the other babies grow. Raspberries spread underground like weeds. They will get all over the place if you let them. They are very vigorous, very hardy plants, difficult to kill. Even in shade or without much water or poor soil they will grow and spread like nothing else.