We really enjoyed hosting you! I'm glad you got to check out some of the must see places.
Yes, sadly, we don't have trains or rapid transit here. Other than SoCo or places up north like The Domain, we are not a "walk around" city like Boston, Dallas, New York or some of the other large cities you might be used to. Even though we are a capitol city, we are a small town that is very spread out, so the bus is all we have if you don't have your own vehicle. It's more like LA here as far as that goes.
When you are a resident of a big city with an underground transit that supports the movement of massive amounts of people quickly, it can seem like our 10 mile location is far from downtown. Especially since the price of an Uber/Lyft has doubled in the last 12 months! I really felt in in LA last when the same ride that cost me $5-7 last year costs $9-$14 last month. I think your suggestion of a car or arranging rides with your host are going to be the best suggestions for Austin no matter which part of town you stay in out here!