I will be spending a few days visiting a friend in Houston, but
really would like to see Austin if that is possible on one of the days when she'll be working. First, is it possible to go for a day, and secondly, since I won't be renting a car, I was wondering if there is bus service from the two cities, such as a greyhound?

There is and I think Greyhound has nonstops, but check their web site. If you're trying to make it a day trip, bus would probably take about 3 hrs. Try looking at SW or CO. Advance fares are usually pretty reasonable (often on sale), many flights a day, and flight takes about 40 min.

Slightly clarification. The 5 daily Greyhounds are direct. Not non-stop. They vary from 1 to 5 stops. [Also, seems like 4 buses use US290, 1 use I-10/TX71.]

If you can, I would probably fly. Like dfr said, you can get some really good deals. The Oasis is a fun spot for dinner and drinks in Austin; it has a gorgeous view of Lake Travis. Hula Hut is fun too.

Southwest roundtrip HOU (Hobby Airport) - AUS from $77.3. ~7 flights each way on weekdays.

Continental roundtrip IAH (Intercontinental) - AUS from $88.3. ~11 flights each way on weekdays.

Thanks everyone, for your great information. Since I'll only be in Houston for a few days, I don't want to get on and off planes, so I think I'll skip Austin this time around and go back later. But now I know about the flights and other options so many thanks!