Personal Disclaimer: I believe that graphic stories and graphic novels would be better terms for this medium, but recognise that the term comic books is familiar to most people, so I'll use that term here.
When did I start reading comic books? I remember the first time I read a comic book was in November 1978, when I was eight years old. That was when comics could still be found in grocery, drug, or convenience stores in circular racks by the doors, which made them enticing for any children entering the store.
I'm not sure what made me ask my maternal grandmother if she would buy some comics for me that first time, but if I had to speculate I'd say it was because most of the comics in the rack were from DC Comics and I recognised some of the characters from the then-popular Super Friends show on television. Whatever the reason, my grandmother proved an easy mark for a pleading grandchild.
The very first comic books I read in November 1978 were DC Comics Presents #6, Detective Comics #482, Justice League of America #163, World's Finest Comics #255, The Brave and the Bold #147, and Super Friends #17.
I didn't read my first book from Marvel Comics until the next month, Avengers #181, and I didn't read my first X-Men issue until April 1980, Uncanny X-Men #135.
By spring of 1979, my grandmother was taking me once a week to a local used bookstore which also sold comic books. The first comic I remember her purchasing for me there was Famous First Editions #7, a 1975 tabloid-sized reprint of All-Star Comics #3 from Winter 1940.
I've continued to read and collect comics since then, with the exception of a period in the 1990s when I couldn't afford the hobby. As of September 2008, I have approximately 6,000 comics in my collection, a number that grows weekly as I continue to buy new releases. Most of my current reading is still centered around DC Comics and Marvel Comics.
These are the items of my collection which are the oldest and/or most valuable: Grand Slam Three Aces #45 (1945), Patsy Walker #22 (1949), Super Magician Comics #12 (1945), True Comics #25 (1943), X-Men #15 (1965), X-Men #27 (1966), X-Men #37 (1967), X-Men #38 (1967), X-Men #39 (1967), X-Men #40 (1968), X-Men #42 (1968), X-Men #45 (1968), X-Men #46 (1968), X-Men #47 (1968), X-Men #48 (1968), and X-Men #101 (1976).
Links to sites about one of my favorite characters (and source of some of my online nicknames): Jean Grey (Wikipedia), Jean Grey and Phoenix, Jean Grey - The First X-Woman.