It's not going to matter all that often, but...
If you really want to know the volume of a pot or pan, start by looking at the underside, near the center, under a bright light -- it's often printed there. If not, just start adding water from a full quart measuring cup. You will soon see how many quarts it holds.
If you're baking something -- a cake, brownies, whatever -- then what really matters is the depth of the material. Most baking dishes are fairly standardized: 8" square, 9" round, and so on, and most recipes will use sizing of this sort. If that's not the case, you will need to guess, based on what sort of thing you're baking and about how deep in the pan you sort of think it ought to be, bearing in mind that most baked goods will rise at least a little bit. If, after pouring the stuff in, it seems deeper than it should be, test for doneness extra-carefully, and expect to bake it a bit long; if it seems shallow, start checking for doneness early and expect it to bake quickly. Don't panic: if you check carefully, what's the worst that's likely to happen?
If you're talking about braising or boiling or something like that in a casserole or saucepan, then there are really two factors: surface area and depth.
If the surface area is too small, obviously, the stuff won't fit. If it's much too large, you may get some scorching in the open spaces, and if there's a lot of liquid it may reduce very fast. Just keep an eye on it, and add a little water if it seems to be drying out.
For covered things, if the depth is too low, obviously the stuff won't fit. If it's much too large, you will get some dead space at the top that won't heat fast, and this may make your food cook a bit slowly or unevenly.
Rule of thumb: For stovetop uncovered cooking, the food should fit with some space around the sides, and if there are many pieces, they each need space (for example, pan-fried chicken pieces should not be pressed up against one another). For covered pot cooking, the food should fit snugly (without actually jamming), so pick the smallest saucepan or whatever into which it will all fit easily. This rule of thumb will take you a very long way. Don't agonize about it: if the pan is too small, you'll have to cook some things in batches, and if it's too large, you just have to pay closer attention to be sure that everything is heating evenly and not scorching. But it's not as though if you have the perfect pan, you don't have to pay attention!
If you guess at the pot size, and keep more or less to this basic rule, the odds of a disaster are pretty slight. If it goes badly, next time you'll know better. This is not something to get wound up about.