Finish: Your paper finish will affect how you're able to print and how much ink your invitation will take. Glossy papers take a lot more ink and longer to dry. Linen papers and papers with more texture have a much higher tendency to bleed, and so they won't work with some at_home printers where you can't change your ink settings. If you're not interested in researching finishes because you're not a weird nerd (like myself) who likes to feel on paper, a matte card stock is an easy, relatively foolproof option that will more than likely work for the printer you're using.
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.