There are several places in the Austin area that rent RVs. Here is one:
http://www.americanleisurerentals.com/index.htm
There is also one near San Antonio (Boerne TX) - it's probably the biggest operation around.
http://www.txrvrent.com/location.asp?ID=1
There are several State Parks in the Hill Country that have camping facilities. Many are on rivers. A few are at lakes. Inks Lake, Pedernales Falls, Guadalupe River, Lost Maples, Garner, - are several very popular camping state parks near Austin. McKinney Falls SP is actually in Austin. All these State Parks are suitable for children.
http://www.tpwd.state.tx.us/spdest/f.../hill_country/
Lake or river swimming will be too cold in March - but maybe not for you Europeans. Probably to cold for a 2 year old.
Audrey