There are lots of things to do.
You could join Kindermusic (my dd loves this), or a mommy and me swim lessons (we like Emler's) for stuff you pay for. I also like stroller strides for a workout for me -- and interaction with other kids for her.
For events that are going on around town -- check out http://metroplexbaby.com/metromoms.htm -- click on the scoop directory and calendar -- and you can see what is going on around town. Many things are cheap or free. Also, you can sign up for their weekly emails to get the calendar sent to your email box. They also host events themselves that are fun.
You could take her to the park - some parks have baby swings. My dd loves the swings.
We like the pet store at the mall.
And the zoo. (Get a membership -- at the dallas zoo it is tax deductible and totally worth it if you are going to go a few times -- but order it online before you go so you get the free parking).
Or the art museum. They have free days -- but I am not sure when they are. Again if you are going to go any number of times -- look into a membership because it is generally worth it.
At home -- we play kiss fest -- where I yell out kiss fest and then kiss her all over -- and she laughs forever. (You can alter it for tickle fest, etc). I also turn on music and we dance around -- my dd loves the 80s! (I put on something different every time and even listen to music I would not normally listen to).
Hope this helps.