One thing that separates me from a lot of other sports fans, I think, is that I have no desire to obtain autographs or personally converse with players. For one, I tend to be a very private person (and thus don't like invading the space of others, either), and secondly (as I've explained in other posts) I like to see the "finished product" of baseball--the actual game--much more than any other part of the bargain. "Don't pay any attention to the man behind the curtain" and all that.
So, when I want to have a different experience at the ballpark, one of my favorite things to do is just get in to watch batting practice. I love to watch the players milling around, some serious, some joking, just getting themselves prepared for the game at hand. I don't begrudge those who angle for pics or autographs (as for some that personal touch is what it is all about), but for me it is enough to just sit back and watch it all happen.
I've had two such experiences, both in roughly the same capacity, that were pretty cool...
At Kauffman Stadium (one of my favorites in all of baseball!), they have a special tour package that lets you in an hour before the gates officially open. I stood right behind the dugout for awhile and watched all the Twins players/broadcasters/coaches trickle out for the batting practice process. My brother got a few autographs, while I just found it cool to be watching something that most don't get to see.
Last season, I did a very similar tour at Dodger Stadium. I got to see Joc Pederson, Corey Seager, and Adrian Gonzalez take their swings, while Joe Torre & Nomahhhhh Garciaparra were just roaming around. The Dodgers were playing the Marlins that day, so I was just feet away when Ichiro bounced out of the dugout (yay!) and also when hitting coach Barry Bonds lumbered out (boo!).
So, when it comes to baseball player interactions, those are the kinds of things I really enjoy. Just laying back and watching the boys prepare for another game.