The OP doesn't give any guidelines, so here's a quick rundown(I'll leave bbq off since you said your kids don't like it):
Saeng Thai House: Excellent Thai. Strip mall that it's located in doesn't look like much, inside is much better. Had the pumpkin curry there a few days ago, really excellent.
Dale's Indian Cuisine: Average Indian food. It's buffet at lunch during the week, not sure about weekends. Nothing on the buffet is spicy, could be a plus or minus for you. My favorite a la carte item is Kheema Mutter, a lamb and green pea curry.
Supermercado El Rancho: Mexican grill located inside a grocery store. Any tacos are good, birria is good, avoid the pozole. Lots of interesting tortas(sandwiches).
El Picante: Has both Tex-Mex and authentic Mexican items. Best authentic Mex stuff is on the specials board. Try the tongue in salsa verde if they have it, otherwise anything in salsa verde is excellent, as are the Ensenada/Nayarit style tacos(fish or shrimp). The most accessible/least intimidating authentic Mexican in Greenville for gringos.
El Azador: Authentic Mexican out by the airport. Absolutely killer on the weekends- try the charcoal grilled chicken. Little atmosphere(very dive-y), but very friendly.
GK Cafe: Cafeteria style with two mains and various sides each day. I forget what they have on Saturday but you can call and they'll tell you. The only main I've had that I was less than thrilled with was turkey bbq. All sides have been excellent(especially the yams, for the love of God get the yams), and they do superb cakes. Lemon cake is my favorite.
Venters Grill: Fried chicken, chitlins, chicken pastry, fresh sides, etc. Had the chitlins and chicken pastry a few weeks ago, both very good. Chicken pastry one of the best I've had(although GK's is right up there). Must-try item here is the sweet potato biscuit.
Marabella Old World Pizza. Lots of interesting and very good European-influenced pizzas. I like the Mondello(fresh mozzarella, prosciutto, arugula). The Toscana is really good, too.
Odds and Ends:
A Tavola does pretty good sandwiches and pizzas. CPW's can be good if they're on a decent monthly specials menu.
There's an indoor flea market on N. Greene St. On some Saturdays, there is a Surly Jamaican Chicken Guy who sells really good jerk chicken that he cooks outside on a grill. $5 a quarter last time I had it. He'll be outside around the corner from the front door if he's there.
That's all that comes to mind now. There are some good places that might be on your way into town, depending on where you're coming from.
Supermercado El Rancho
Greenville, NC, Greenville, NC
1218 Mumford Rd, Greenville, NC 27834
Dale's Indian Cuisine
419 Evans St, Greenville, NC 27858
2422 Stanton Sq, Greenville, NC 27834
Anchalee Thai Restaurant
3400 S Memorial Dr Ste 17, Greenville, NC 27834
You're welcome, and welcome to the area. Greenville is a tough, but not impossible, town for non-chain food. Several of the smaller nearby cities like Wilson and Kinston have a lot of locally owned restaurants(mostly in the taqueria or greasy spoon category, Chef and the Farmer in Kinston being a notable exception).
The main thing that the above post lacks is coverage of barbecue, which I left off due to the requirements of the OP. A search should turn up some good information on Skylight Inn(Pete Jones'), B's, and Blackbeard's.
Thanks! We tried Chef and the Farmer. It was good, but I think we'd heard so much about it that it was impossible for it to live up the hype. I'm sure we'll go back though.
I saw in a search that Japan Inn was the best place for sushi. Is that still the case? Also, is there a good bakery? Or is Fresh Market the best bet?
Japan Inn is still the best, I'd say. Sappari is okay, I don't really like the others.
Fresh Market is your best bet for bread. There is Swiss Chalet down by campus, but IIRC they mostly do pastries. There is also a Mexican bakery called La Ideal in the shopping center behind the Burger King on Memorial Dr., near Pitt Community College. They pastries and sweets, and have great prices- most everything is 75 cents or so per item. They also have really good tres leches cake in one of the coolers.