I grew up reading newspaper comics, and my two biggest favorites were Charles Schulz's Peanuts and Bill Watterson's Calvin and Hobbes. I also enjoyed Garfield, The Far Side, and FoxTrot (I miss when Bill Amend did daily strips sometimes).
I discovered webcomics just before I hit college. Sluggy Freelance was my gateway, but of the dozens I read, Megatokyo was my biggest influence. The pencil artwork is really good, plus I learned panel layouts and not having to end every page with a punchline so long as the story keeps moving.
Kiyohiko Azuma's works (Azumanga Daioh, Yotsuba&!) never cease to amuse me how they find humor in the mundane and everyday.