Madison is a very veg-friendly city. As a liberal, environmentally conscious city with one of the biggest and best Farmer's Markets in the country, Madison is a great place to buy and eat fresh vegetables! Many local restaurants consistently use fresh, local ingredients. And if you eat dairy, you could come to no better place. The dairy in Wisconsin is like none other.
A great, exclusively-vegetarian restaurant I've been to is Green Owl Cafe, http://thegreenowlcafe.com/, which has delicious vegetarian and vegan versions of everything you can eat with meat, from spaghetti and "meatballs" to the their vegan bloody mary. Their menu includes American, Mediterranean, German, Indian, and Mexican-style food, with no meat in sight. It is on Atwood Ave, in the Near East, which is an approx 10 to 15-min bus ride from downtown/campus.
Another exclusively-vegetarian spot to grab a bite is Mother Fool's Coffee House, an artsy cafe also in the Near East on Williamson Street. Probably a 10-min bus ride.
http://www.motherfools.com/. I've never been there myself, but it's a local favorite.
Another great option is Monty's Blue Plate Diner. A local favorite, it is a traditional diner-style restaurant with a wide variety of traditional American and unique dishes that provides a vegetarian and vegan option for almost every item on the menu. It is also on Atwood Ave, near Green Owl. http://www.montysblueplatediner.com/
Here is a list of vegetarian and veg-friendly restaurants and grocery stores in and near downtown:
http://www.vegmadiso...estaurants.html
Of these, Alchemy's, Bunky's, Dobhan, (all on Atwood, and delicious!) and Bandung are Near East.
If you are in the campus/downtown area (campus and downtown are adjacent to one another), there is a plethora of restaurants on State Street and near the capitol, many of which are veg-friendly. Look for East Asian restaurants like Rising Suns (delicious thai restaurant on state), Tibetan food like Himal Chuli, and Ethiopian restaurant Buraka for great vegetarian options. State Street Teahouse Dobra Tea is vegetarian-only. Although ethnic, especially Asian, restaurants are likely to offer many vegetarian options, most Western food restaurants in the area are vegetarian-sensitive and offer plenty of veg options.
Madison also has several delicious Indian restaurants, which are of course great for vegetarians. In the downtown/capital area, Mirch Masala
(haven't eaten there myself) is on the Capitol Square, and the absolutely delicious Maharani (one of my all-time Madison favorites) http://www.maharanimadison.com/ is on W Wash just two blocks from the Capitol Square.
As far as cleanliness goes, you don't have to worry. Madison has an excellent, diverse, and reputable restaurant scene, and the restaurants in the central area are all clean and well-maintained.
Now as far as living west of the Capitol, that is a good location. It's on Lake Monona, which is beautiful. Many students live there, but it is not rowdy like some student areas. Anything near the Capitol is an excellent place to live, since you have access to campus, downtown, Capitol Square, restaurants, shops, and all events and festivals going on downtown.
I'm not sure if I understood you correctly, but I gather you would be at the College of Agricultural and Life Sciences. You're not going to be able to walk there from Near East or W of the Capitol to there in less than 1/2 an hour, probably. However, it is totally easy to take the bus! Both inter and intra-campus buses come regularly and many different buses go between campus and the more central downtown area, as well as to the Near East.
Here is the bus website: http://www.cityofmadison.com/metro/ (or you can use Google Maps to plan your bus trip). As a student, you'll get a free bus pass which you'll have to renew about every semester.
You'll be able to take the bus from west of the Capitol in about 10 mins, and from the Near East in maybe 20 mins, depending on where in the Near East you live (closer to Campus like on E Johnson, or further away like Willy St or Atwood Ave.)
Also, you should consider a bike, if you enjoy biking. Madison is a great biking city.
I think that if you want a truly walkable distance to the west side of campus, where it seems like you will be, I would look for apartments in the Regent Street/ Camp Randall Stadium area. It is not a bad area for students to live; it is right next to the Monroe St/ Vilas neighborhood area, which is a charming and quiet residental neighborhood with lots of shops and cafes, a nice lake, and a Trader Joe's grocery store.
Let me know if you have any more questions I can answer!
Hope that helps.
