The Peppermill is the classiest place in Reno these days, IMO.
But you will need a taxi to get there.

Multiple dining options, and the casino & property is really nice.

I was in Reno last summer, and it's changed a lot since 20 years ago. Reno is showing its age, unfortunately.

For walking distance from Harrah's, check out the Eldorado. They always had good Italian food, with the Carano family's heritage. It's been a while since I was there however.