Surprised no one has suggested a visit to the Lady Bird Wildflower Center. At the end of April, you will be there very near the peak season. We like to combine a trip to the Wildflower Center with a trip to the Saltlick BBQ in Driftwood. Both are south of Austin. Even my quasi-vegetarian elderly parents love the Saltlick and we've been taking my daughter there since she was a week old.
Hamilton Pool is one of our favorites. It's a pond lying under a caved-in cave. Hard to describe, but very beautiful and well worth the 45 minutes from where we live in central Austin. You can even walk to the Pedernales River from there.
Mt Bonnell is accessible via an easy five minute walk to the summit. Definitely nice views and close to the center of town.
Take your toddler to Zilker Park. He will love the Zilker Zephyr, a train that will take you around a small part of the park. You can combine this visit with a trip to the Botanical Gardens also in Zilker. The gardens include a lovely Japanese Garden. There is also a lovely walk along the river (called Town Lake) where you can watch the turtles and feed the ducks. There's even a large playground in Zilker Park. April is the best time to go.
If it rains, visit the LBJ Library on UT's campus. The temporary exhibitions are usually very good. The permanent exhibition showing a various assortment of over-the-top gifts from heads of state is well worth it. And the LBJ is free.