The first thing to know about getting around in Austin is that Loop 1 and Mo-Pac Blvd are the same road. I guess the city and the state had a disagreement long ago about what to call it.
My personal idea of a bike holiday in Austin would be easy riding, minimum traffic issues, and lots of eccentric little places where you can soak up the old Austin vibe. My plan misses out on all the beautiful hill country scenery.
The big hills are mainly to the west of Loop 1. Smaller hills are south of the river. The area bounded by Loop 1, highway 183, and the river is mostly flat and has the old style street grid layout with many alternate routes.
Downtown between the University of Texas and the river is fun to explore on a bike. You'll see lots of other bikes there. You have to watch the traffic of course but the downtown motorists are highly aware of bikes. There are some great bike bridges that let you get to the urban park areas south of the river. Personally I like the Pfluger bike bridge just east of Lamar, and the South First Street bike bridge.
If you want to do an easy all-day urban bike ride with many possibilities for stopping to explore or refresh, ride the length of Bike Route 31 from Research Blvd (north side of town) to William Cannon (south side of town). Route 31 has a few interruptions and zigzags you'll have to deal with.
If you're so inclined, you can learn a bit about the Capital Metro system here. All the buses have racks for two bikes. If you get tired or encounter any trouble, you can easily catch a bus for a boost or to bail out. The best deal is usually to buy 24-hour passes as you board the bus.
If you want some hills and scenery, ride west on either Far West Blvd or Steck. There's a nice bike/ped bridge over Loop 1 at Far West which connects to Shoal Creek Blvd through a little park. Shoal Creek is part of Bike Route 31.
Whatever you decide to do, enjoy the vacation here.