AT_HOME PRINTING: By far the most accessible, this is easiest option for tackling the project in smaller chunks on weeknights rather than spending hours in a studio or waiting at a print shop. However, you should test paper before buying a ton of it, printers can be especially finicky, and your can't exactly change the ink your printer uses. If your printer isn't higher end, you may have very limited paper options. Here's a little more on at_home printing.
Weight: Paper thickness is measured by weight. The higher number you see, in pounds, on the packaging label, the thicker your paper is. This doesn't mean you have to find the thickest paper you can—most card stocks are a great weight for invitations and can hold up to being mailed.