You're not going to find a hotel room downtown for under $300/night over MSU weekend and, personally, I think that's asking a little much for a hotel in Knoxville unless you pony up even more for the Oliver/Hyatt/Tennessean; the Hilton is a fine hotel and I'm a big fan of that Hampton Inn, but asking those prices is rather egregious.
Here's a link to downtown hotels and their cost/night for that weekend.
If at all possible, however, I would recommend staying downtown. Most parts of Knoxville are great, but there's something to be said about being able to experience downtown, the Old City and campus within walking distance while being a $10 Uber away from the "cool" parts of North Knoxville (Happy Holler/Old North/4th and Gill).
Here's a downtown Airbnb that totals out to $550 for two nights, which is excellent all things considered as you won't find any hotel nearly as nice for $275/night all-in (hard to beat a rooftop on Market Square, truly). Keep in mind that staying downtown will defray your transportation costs significantly, as it'll be cheaper to pay $7/night to park and walk nearly everywhere than Ubering from 20 miles away or driving yourself and hoping to find parking/figuring out how to make it home post-game.
If you decide that staying in West Knoxville is better, here's a list of hotel availability that weekend; I'd definitely go with Home2 and think it's a truly great option, though I still feel people your age would be better suited downtown than in Anywhere, USA for your first visit to Knoxville.
If I were you, I'd plan on checking in to wherever on Friday night and look to get dinner around downtown. Lonesome Dove is great if you're into pricey steak and wild game, while Cafe 4 is quality American cuisine in a setting that would surely appeal to your wife; if you want a more laid back Knoxville experience, Downtown Grill & Brewery has great food right next to Market Square and Barley's in the Old City is a classic for pizza and beer. There are plenty of bars in Market Square (Pres Pub for live music w/ a rooftop and smoking on the first floor, Scruffy City for a less-crowded brother to PresPub, Peter Kern's Library for cocktails at a "speakeasy"), general downtown (Sapphire for an "upscale" cocktail lounge with $3 PBR, Suttree's for some excellent draft beer selections, Clancy's for a (hopefully) quiet Irish pub) and the Old City (Southbound/Hanna's on the off chance you want a terrible club, Urban Bar for a wonderful smokey dive, Central Depot for just a great bar all-around) from which to choose, or you could go bowling at Maple Hall/see a show at the Tennessee Theatre or the Bijou. On Saturday, I think that you'd enjoy checking out one of the few remaining bars on The Strip right next to campus (Half Barrel, less-clubby Hanna's, Cool Beans, maybe even the Fort Sanders Yacht Club) before tailgating unless you can luck into getting a seat at Calhoun's on the River before heading over to campus. Win or lose, you'll definitely find something to do Saturday night either downtown or in one of the adjacent neighborhoods (there are plenty of breweries/small bars in the 4th and Gill/Happy Holler areas).
Enjoy your trip!
