Unfortunately most everybody in Richmond drives. Richmond is doable without a car, but imho just barely. There is a public bus service (free with student ID I think) http://www.ridegrtc.com/
and it recently started coming to campus, but it only serves a limited area without connecting to another bus (which tends to be a pain). Fortunately the area its serves has some of the best housing options. The route 16 bus runs from campus east down Grove ave.
Anywhere within a couple of blocks of Grove is nice and very safe, but pricey and a little hard to find rentals. As you get further from campus toward "museum district" and "the fan" there are a lot more options and while it’s not quite as safe as you move farther east, it’s still safe enough and there is more to do: walkable grocery stores, restaurants, bars, coffee shops, parks etc. "The museum district" is about 10 minutes bus ride from campus and a little more residential while the fan is just about 12 min. and a little more happening.
Those two areas are where I would look to take advantage of the bus, but keep in mind buses don't always run on time and stop running some routes in the middle of the day and at night. UR has a shuttle that supplements the city buses but it also doesn't run 24-7. Think the last run is 1030ish. There are lots of other areas with a more suburban feel and more typical apartment complexes, but if you want to live in Richmond and not just sprawl (see short pump) consider these two areas. Sorry to say it but you will likely need a car to survive UR, but with a little patience and planning you could minimize car trips.
The city itself is generally bike friendly and becoming more so, but the area immediately around campus is hard to navigate and honestly a little dangerous for bikes (dark & hilly). The buses do have bike racks so maybe that combination would help. Hope this helps.