Should I let the chili mature on the plant, cut it open and take the seeds, dry them and store in a cold, dark and dry place? Or is there more to it than that.
You've pretty much got the idea; some details to add:
Seeds are ready when the fruit is mature. Make sure they're ripe and fully colored. Dry away from sunlight -- and make sure they're fully dry. Test dryness by bending a seed: if it snaps in half, it's ready. If it bends, it's not dry enough. Store in a cool, dark, dry place. Use within 2-3 years.
Cross-pollination is an issue: isolate from other varieties by 500' or cage plants to ensure pure strains.
For long-term success (i.e. good results over multiple generations), you'll want to save seed from at least 20 plants. If you don't do this, the gene pool doesn't have enough variety and you'll end up with poor plants after several generations. If you only want to save seed for a generation or two, this probably won't be so much of an issue. (If you save seed from a couple of chilis in generation 1, you will probably have enough seeds for 2-3 years at which point you can buy professionally grown seed again to refresh the gene pool.)
(Source for details above: "Seed to Seed" by Suzanne Ashworth.)