We have a lot of East-side favorites, all local, or at least local chains (the Food Fight chain is all in Dane County, for example).
Monty's Blue Plate Diner on Atwood. Great unpretentious food, not too expensive, very comfortable atmosphere. Their website is here:
Monty's Blue Plate Diner | A Neighborhood Favorite
Wong's Garden on Atwood. Not more than about 15 tables in the place, owned and run by a warmly welcoming Chinese family, serves really, really good Chinese food in a simple atmosphere.
Bunky's, also on Atwood almost to Fair Oaks, also comes to mind. It's a little pricier - entrees are in the $10-$18 range - but excellent Italian and Moroccan food. Their Chicken Shawarma is heavenly. Owned by the daughter of the original Bunky (whose Italian restaurant was a fixture in the old Greenbush neighborhood) and her husband.
Rossario's on Monona Drive (which is actually the same street as Atwood, the name just changes once you get into Monona). Very, very good, simple Italian food, very friendly owner and staff, just a really comfortable place to eat. Spouse and I each had soup, entree and a glass of wine there on Friday and the bill was around $30.
El Pescador on East Washington, just across Johnson Street from Ella's Deli. Excellent Mexican seafood place, very authentic. It can be a bit spicy for midwestern tastebuds, but is very much worth trying.
I agree with Bowie about Ella's, by the way. Good Kosher-style deli food (the only place in town where Spouse can satisfy his craving for chopped liver and sweet & sour cabbage soup) and the inside must be seen to be believed. Kids of all ages (like my spouse) have a wonderful time playing there.
Skip Zach's Avenue Bar on East Wash, closer in toward downtown. Classic Wisconsin-style bar/restaurant, famous fish fry and fish boil, good basic meals with lots of food for not much money. And you'd be astounded at the number of Capitol Hill movers and shakers you'll see there. Expect crowds and a long wait on Friday nights, but it's worth waiting for.
Roman Candle Pizza at 1054 Willy Street. Excellent thin-crust pizza, very good soups, decent pasta, surprisingly good wine and beer selection.
Glass Nickel Pizza on Atwood at the corner of Fair Oaks. Excellent thicker-crust pizza, very good salads, very good pasta, lots of beers to choose from. Occasionally has live music on weekends.
That should get you started - have fun, and let us know which you like!