On 57th:
The Medici Bakery has some of the best breads in the city. Baguettes and multi-grain are particularly good.
In the back of University Market, the deli counter makes very good sandwiches using Medici bread. Owner Hans is especially proud of the liver sausage.
On 55th:
Bonjour Bakery makes some good pastries (I'm partial to the morning roll and the lemon-pistachio cookies) and some pretty good French-style sandwiches. I enjoy their house-roasted Sumatra coffee.
On 53rd:
Harold's Chicken Shack should be tried by everyone. I'd suggest a half with hot and mild (plus plenty of cold beer). Not the best Harold's but it gives a pretty good idea.
Nathan's is basically a hot dog stand you wouldn't look twice at but they have a separate Jamaican menu. There's much better to be had outside of Hyde Park though; try Tropic Island on 79th.
Rajun Cajun is your only one-stop source for Indian home cooking and soul food. Nothing spectacular but can be enjoyable. The stew-like offerings are often the best bet (they survive on the steam table).
Valois is a classic steam table cafeteria dating from 1921 (recently remodeled). Good breakfasts and things like short ribs.
Dixie Kitchen (Harper Court) has fairly decent Southern cooking. The gratis johnnycakes may be the best thing there.
On 51st:
Original Pancake House is part of a national chain but serves really excellent breakfasts. Try the apple pancake if you're hungry.
Within a few miles of Hyde Park there is a wealth of good food that will keep you busy exploring for years.