Generally, east of I-35 is the less attractive part of town, and definitely stay out of the midtown area east of I-35 after dark. It's not all bad, but you wouldn't know as a visitor what's okay and what's not.
Texas (and Austin) has a ton of publications of things to do and places to see. If you're driving, the welcome stations at highway entries have them. If you're flying in and renting a car, browse through the Texas dot-gov website.
The Colorado River is impounded by a series of dams, including a small one which makes "Town Lake" in the center of twon. Lake Austin is almost downtown, what with the city's growth in recent decades.
Austin is a sprawling city. For sure, buy a street map. Much of the city is NOT a grid system, once you're away from the original pre-WW II part.
There is a free counter-culture newspaper which lists clubs' musical events and gives a lot of other entertainment information. I'm blanked on the name at the moment. The local paper, the Austin American-Statesman, also has a lot of info.
Lots of little esoteric bits: The French Embassy from Texas' Republic days, east a couple of blocks from I-35 around the 1400 block. Or, out the Bull Creek Road (FM or RR 2222) is a pretty drive. If it's still the same name, a BBQ joint called the "Salt Lick" (IIRC) used to be good. Just past it is a little road to the right. A short ways up on the left, you can see in the limestone bottom of Bull Creek some grooves from the old wagon wheels where people used the creek as a highway to get to town.
If you like to look at older, beautiful homes, the area west of Lamar Blvd between Enfield Road and W. 24th St. has a goodly number. On west, if you take Enfield Road to Lake Austin, you'll find some good restaurants. Scenic Drive winds along the lake, mostly above it on a cliff.
"Austin ain't Austin, anymore--but it's still the best city there is." -- Jerry Jeff Walker
, Art