If you're buying your own paper, you can find a good selection at local office shops when it comes to card stocks and glossy photo paper. However, if you're looking for linens or specialty papers, you may have to go online. Always get a sample before buying a huge stack of paper, and do a test print, to make sure that whoever or where ever you're printing from can actually use the paper you've selected with your design and medium. Here are a few other places to buy paper from.
Letterpress: The designer and paper lover's dream, letterpress is visually lovely and highly coveted all over Pinterest. However, it is expensive because it is a fine art that requires some heavy machinery. Letterpress is the option to use when you want to support the artists making it, not because you think you should. Thomas Printers is a great option for people looking for letterpress invites on a reasonable budget. The other option to letterpress is to build your own, but you better have a serious love for the actual craft. The majority of your guests, sadly, won't know or won't care that your invites are letterpressed.