I don't know that I could name a single favorite, but I have some fun memories.
The first autograph I ever got was Joe Charbonneau on a free Xeroxed photo at a card show in 1991. I had just turned 7 years old. The second one was Mark Lewis (in ballpoint pen on a roster insert from a game program) a couple months later. The third one was a random rookie Blue Jays pitcher on the back of my ticket envelope in September that year. That rookie went on to win a Cy Young Award five years later-- Pat Hentgen. I still have all three of those sigs.
I didn't do a whole lot with autographs for the next few years. If a guy was signing free at a card show, my dad and I would get him-- Carlos Baerga, Ozzie Newsome, Bobby Phills, Hanford Dixon, Eddie Johnson, Derek Lilliquist, Kevin Mack, Dick Ambrose, Bob Gain, Otto Orf, Mel Harder, Alex Delvecchio. My grandmother would keep an eye out for shows in Michigan as well; she got me former Tigers like Mark Fidrych and Denny McLain.
I didn't get back into the hobby heavily until college. I mailed out a few in the years between them-- I remember Sandy Alomar Jr. was the first one I ever got back; this would have been 1992 or 1993. I started in again with hockey in the late 90's-- Trevor Linden, Alexander Mogilny, Dave Andreychuk, Mario Lemieux, Ron Francis, Jaromir Jagr, Pavel Bure, Brett Hull, Marty McSorley, Nikolai Khabibulin, Gordie Howe, Maurice and Henri Richard, Guy Lafleur, Patrick Roy... I tracked everything in a notebook. I wish I still had it; it would be fun to add all of those to the spreadsheet I have now.
In college (this is 2003 we're at now), I got into it because I was finally in a hockey city. Growing up near Cleveland, we didn't have much puck aside from the Lumberjacks and Barons. But now I was finally in the middle of everything. Over Christmas, I searched my cards for guys who were still playing and on my first day back, I trekked out to the FleetCenter (as it was then known) and waited. And had no luck aside from Hal Gill as he walked in from his North End apartment. I remember going on Paul Buxton's site (some of you may know him as an early hockey 'graphing guru) and getting in contact with him via AIM (like I said, 2003), getting advice from him and adding myself to his Traders' Bazaar, where I found a few other good traders. I eventually got to know a few other collectors around Boston-- notably Lanny (crosbyfan on a number of boards; he now lives near Edmonton, I believe) who was just getting into it at the same time I was. Pretty quickly we figured out which hotels to try and when the Canadiens came to town a few weeks later, I was off and rolling.
I actually posted a bunch of my recaps from my old college site on my blog a couple years ago. I'll need to post more of those soon. Check them out as I think my optimistic 19-year old voice from in the moment captures it better than today's faulty memory at a point 10+ years into the future from a cynical 31-year old:
http://dfwautographs.blogspot.com/2013/08/hounding-report-rangers-tunnel-august.html
Obviously now, I'm still pretty serious about it. I've sent out over 750 requests since 2007, and since moving to the Dallas-Fort Worth area I get out to multiple MLB games, minor league ballgames, card shows, and NHL practices every year. I don't do the hotels much, if ever, but I still get stuff done as often as I can.