1) Until 1954, most players left their gloves on the field while their team was at bat.
2) Relief pitchers used to ride in little cars from the bullpen to the pitcher's mound.
You probably already knew about those, but here's one I just discovered:
3) When a pitcher struck out to end the inning, the catcher would immediately toss the ball to him. It happens twice in this 20-second span of video:
It's easy enough to figure out why this practice developed, but I have no idea when it started, and a cursory Google search turned up nothing. I figure it ended when batting gloves became de rigueur, forcing pitchers to make a pit stop at the dugout before heading back to the mound. Or maybe it fell out of habit after the DH was adopted, and half the pitchers stopped batting altogether. But those are just guesses. Anyone have any better ideas?