I caught the Tigers and Indians at Progressive field in Cleveland a few weeks ago and I made sure to check ahead of time about camera policy. I was able to bring it in no problem. Their basic policy is it’s fine, so long as you’re not being disruptive to other spectators. But, it does seem to vary by location. I’ve heard of places saying you can take photos, but “cameras with removable lenses” are not allowed.
For a first try at such a thing, and considering the fact I have only entry-level gear that is not very well-suited to such situations, I was pretty satisfied with the results. For most of the shots I used the less-than-stellar Canon 55-250mm lens. I had to crank the ISO up to 3200 for most shots to get decent shutter speeds but the bright colors on the field made the noise less noticeable. I didn’t even bother with RAW.
For shots in the stands of friends and family, I’d forget about bringing along the big external flash, as you have no good way to capitalize on what it has to offer. Bump your ISO, and just use the stadium lights as your main light source. If necessary, use some subtle pop-up flash to fill in face shadows, or make your subjects pop a bit more. You’re probably going to want to be well below your max sync-speed to get a nice background exposure and establish context. You can then tweak the exposure of the subjects with manual flash or FEC, until you have a nice result. You should be able to dial in the flash such that your subjects are well balanced with the background but their faces don’t have that signature pop-up flash “nuked” appearance. Depending on where you’re standing relative to the stadium lights, the flash might be altogether unnecessary.