First off; Welcome to DIA and more so. Denver, Colorado. I'm just outside of DIA in the nearby Denver suburbia of Aurora.
Unfortunately, there isn't much in walking distance from DIA that I know of. I rarely find myself over there (despite it being almost literally down the road.) However if you had a car I would suggest a Beni Hana or a couple of Bars and restaurants that are a few driving miles away in Aurora here. I believe you can catch an Airport shuttle To and from The Airport area and we do have a running Lightrail system (Much like a subway, except above ground.) that runs through denver and stops fairly close to the DIA shuttle grounds.
A good ordeal is to open up Google maps again and look for businesses.
In what put "Food" in where put DIA (or Aurora, Colorado) and you should get some search results.
The Aurora mall is just off of I-225 (west of DIA) between 225 and Chambers. There's plenty to eat and do there on an afternoon, and just across the street from the mall is the Aurora Town Center which has quite a few stores to keep you occupied during the day. All of these can be accessed easily by Taxi on a fairly decent fair if you've got some extra cash on hand.
HOWEVER! Thats just my prowling grounds which I know best.
and another place you would really be interested in if you can hop a ride would be Southlands. It's another shopping center that's actually fairly new here and pretty sprucy with a good number of stores ranging from barnes and Noble to a great Movie theater.
Southlands is just south of DIA if you take 470 south.
Here's some Mapquest information if you're interested.
http://www.mapquest.com/directions/main.…
DIA -> Southlands Shopping Center
http://www.mapquest.com/directions/main.…
DIA -> The Aurora Mall / Town Center
The Aurora Town Center and mall is a bit further, so I would really suggest Southlands unless you can find a good cheap ride. Besides, Southlands is prettier in my opinion.
Let me know if you need anything else!