Visit any craft store and you can easily become overwhelmed by the hundreds of papers available for purchase. It's all too easy to become addicted to obtaining everything presented, so I'm going to list the papers I always keep in my inventory. An assortment such as this can easily be all that's required for the majority of your projects.
Plain Quality Cardstock in assorted colors: A basic necessity. I prefer a 110# weight. This is heavy enough to make your own base cards. Why spend $5.00 for 25 precut cards when you can purchase 150 sheets of white cardstock for less than $10.00. It's also used for die cuts, frames, backgrounds, almost anything you can think of. I often print on it.
Patterned Scrapbook Paper: Another required supply. These can be purchased as single sheets or in stacks. Extremely reasonable when purchased in stacks with a 40% off coupon from your local craft store. They come in an unlimited number of themes and colors.
Textured Cardstock: Really important for giving your cards or scrapbook pages a special look. They come in plain colors and in patterns, and often give the impression of fabric rather than paper. I prefer the small 4 1/2" x 6 1/2" stacks because they only need a little trimming to fit perfectly on a card.
Metallic Cardstock: Gives a gorgeous, elegant look to any card or scrapbook page. You can also get matching colored envelopes.
Velum: Use it as an overlay, or for printing.
Parchment Paper: I line all my cards on both interior sides with parchment paper. Makes a perfect letter writing medium or a simple site to stamp a sentiment.
Well, there you have my basic essential papers. But just like everyone else, I have my small stash of non-essentials: handmade paper, mulberry paper, metallic-textured cardstock, glossy paper, corrugated cardstock, etc, etc, etc! And yes, I do use them occasionally. But if you stick with the basics I've listed and add a few of your own favorites as you go along, you won't go wrong!