Make sure you hit the zoo really early because it can be really hard on an infant who will be riding in a stroller because of the heat. When my daughters were that age, the simple things seemed to work best. There are so many ways to entertain a little one in our area, here are some of our favorites.
Parks - find one with swings & you're set; try to go early before it gets too hot.
There are several pools that have baby/toddler areas - my girls love water and would stay in the pool forever if I'd let them even when they were a few months old.
Dallas Arboretum - great for wonderful pictures and picnics.
Mall playgrounds (Frisco, Mesquite, ect.) - the are perfect for crawlers because they are free to explore, plus it is air conditioned.
Frontiers of Flight Museum @ Love Field - really neat kids area for playing/crawling, plus my husband loves the exhibits.
Lakes (White Rock, City Lake, ect.) - there are lots of small lakes with ducks in the area and feeding them can be a lot of fun.
Rain Forest Cafe @ Grapevine Mills Mall - my girls love this place, it's like the zoo but air conditioned and the realistic looking animals come alive.
Libraries - most local libraries have infant story times during the week that are fun. They sing songs, read, and some have toy time, too. Or bookstores have them too, but they are usually very crowded and focus on the 3 and older kids.
Farmer's Market - so fun to be outside & let them help pick out & hold fruit.
Check out your citites parks & recreation page for more things to do and information on things they offer like pools, parks, etc. I know it doesn't sound like city events can be fun, but when babies are little they would rather be crawling or splashing or swinging than stuck in a stroller wondering what is in the distance. Also, it is hard to keep them happy, dry, fed, well rested, ect. when they are so little so remember not to expect them to love it as much as you do. I promise you will have so much fun watching her do things, even if they are simple. We took my girls to Galveston this past week with lots of plans to spend several days at Moody Gardens looking at the Rain Forest, Butterflies, Water Park, Aquarium, paddle boat, ect. However, when we got there all my girls wanted to do wat play in the knee high water and roll around & play in the sand! My husband & I were disappointed we missed the other stuff, but we were happy because they had an AWESOME TIME in the water (and we saved $100)! Kid's see stuff all of the time, but it's not very often they get to get hands on and experience things in new places. Wow, this is long - but I hope it helps!