Jannie,
I think going to T's baseball game would feel a little weird to me too. It would feel too much like I was supposed to be his "fan" or something. However, he probably was just being nice and helping you find something to do with your time now that school is out.
Have you thought about volunteering at a local day camp? Maybe you could do arts and crafts with kids or read to them....just a thought.
I would not mind if my T invited me somewhere--but he never has. I would even go with hubby. But, not his softball game...In general, I like it when T self-discloses which he does fairly frequently, talking about his kids or, on occasion, his girlfriend (ugh)....