By staying at the Fairmount, you are in the "East Randolph Street" area of downtown Chicago. It is not convenient to public transportation, although you can get to it by walking at least three blocks. But don't rent a car, for the reasons already stated. Best bet is to take a taxi and pay a 10% tip. You'll find your roundtrip fare to be about $12-14 if you stay within a 2 mile radius.
Your hotel is within walking distance to four good restaurants. First, right in your hotel is Aria. Its chef was a competitor in Top Chef Texas, Ms. Kim. Ask for her and compliment her on her showing on TV and she might do something special for you.
Second, The Palm is right across the street. Chicago is known for its steakhouses. Many of them are chains, but I still think they are very good. The Palm is one, and I have never been disappointed in a meal there. The atmosphere is not stuffy. (FYI, your neighborhood is mostly business and convention traveler heavy; so the style of the nearby restaurants will mirror that.)
Third, Tavern at the Park is a very good restaurant with a varied menu of standard American cuisine. It is also somewhat laid back. It is only three blocks away from your hotel on Randolph Street. Owned by same owner as Keefer's.
Fourth, for lunch, Sweetwater on Michigan Avenue and Lake Street is very popular.
For breakfast, Wildberry on Randolph is also very popular.
Going farther afield, you can't go wrong looking at other recommendations based on your price range, your selection on cuisine, etc. But I would recommend going no farther than 2-3 miles away from your hotel. There is so much in that area that you spare yourself the hassle and greater expense of going to an unfamiliar neighborhood far away. One excellent "restaurant strip" is West Randolph Street at 800 to 1000 West. Many great restaurants to choose from there. River North is also good for a range of gastronomical experiences without the expense and hype of Michigan Avenue/Gold Coast.
Another suggestion for you. The Ritz is only 1 -1/2 block away from Ditka's. This is a steak restaurant with an acceptably diverse menu. Yes, it is in fact owned by Chicago's own Mike Ditka, and he often spends time there. I think the food is very good, and you will definitely get some Chicago ambiance. Yes, its a steak joint frequented by business guys and sports fans, but hey that's Chicago.
A truly outstanding restaurant is also one block away from the Ritz. NOMI in the Park Hyatt. Its eclectic Asian fusion cuisine is outstanding and nearly as pricey as Spiaggia. If you can afford it, it would be a do-not-miss close to your hotel.
Also one block away from you is RL. Fans swear by it, particularly one ex-New Yorker I know.
A spectacular place to eat, with good food, is the Signature Room at the top of the John Hancock Building, across the street from the Ritz. I have found the dinners served at this standard Americanized-continental restaurant to be very good, but the attraction is the incredible view of Chicago from 95 stories up. Only worthwhile if the sky is clear to allow for the view.
Finally, for moderately priced but really great Italian food, Francesca's is right around the corner from the Ritz. Authentic Italian (even pizza, continental style) with a happy crowd and a busy vibe. Piano music, at least on the weekends.