Some people walk/bike and maybe a tiny tiny few take the bus (the bus is a little obnoxious because only one line comes to the vet school and it doesn't come more than 1-2x per hour so if you miss it or if it's running late, it can really screw things up). I'd say 90% of us drive- parking is $660/year, which is frustrating, but the convenience of having a car and being able to live slightly further away (ie lower rent) is invaluable in my opinion.
Two of the close-by places that get a lot of first year students are The Village at Colbert Park and also The Place @ 117. They're apartment complexes within 10ish minutes of the school, pet friendly, decent rent (if you have roommates especially), and do roommate matching so that can be a big plus if you want lower rent but don't know anyone to room with.
Otherwise there are a million apartment complexes nearby given it's a campus town. Also duplexes, condos, etc. Rent in Savoy (slightly south but v close) or Tolono (further south, more of a drive) is usually lower than Champaign. Urbana rents are often also less than Champaign, but certain neighborhoods in Urbana are...less desirable.
Using GoogleMaps to scope out an area is great if you can't make it down here in person, and otherwise you can post here and ask if anyone has experience with a certain landlord, etc!