We don't do these yet, but here are some activities I did when I was a nanny: swimming pool, playgroups, play dates, the park (once the baby was cruising, there were play parks that he could explore). At your baby's age, I cannot really think of anything else at the moment.
Oh, and we take him to Mets games. It was a lot easier last year when he was less mobile, but he loves flirting with the crowds around us! The people behind us were so charmed by him that they bought him a hat!
Story time, play on the swings or slide at the park, swimming, but especially go out to eat or walk around anywhere that there's a lot of poeple to look at. Like the mall or Target. AJ looooooves to people watch. Except for me people watching is discrete and for him it's staring at them with his big blue eyes until they look at him.
play in the pool in the yard, walks in the stroller, go to baby music class. Not much. I'm looking forward to him walking so we can take him places to play. He hates parks, the swings and slides freak him out.
With a non walking child I might go to a baby pool, to a concert, a farms market. You could try a music class. I think they are ideal for sitting babies.
Once they can walk they enjoy stuff like farms (many in long island, nj, ct with family friendly events).
It is fine to just take her with you and do something you'd enjoy if you have a baby that is accommodating. That is one of the big advantages of the very young children.
Oh, and we take him to Mets games. It was a lot easier last year when he was less mobile, but he loves flirting with the crowds around us! The people behind us were so charmed by him that they bought him a hat!