A first time visitor to Cleveland
Hi Lanyboy -- Welcome to CLE!..you should get some great recommendations from the CLE chow board members. But give us some basic info; where will you be staying?, will you have transportation when here?, when are you arriving?, etcetera. It always makes it easier to pinpoint places and suggestions when the above information is known!...and I'm sure you'll receive many great ideas for your upcoming visit!
I wont leave you totally hanging, though -- best corned beef in CLE...Slyman's and it would qualify in the Diners category...if dining in, be prepared for a frantic/manic experience! http://www.slymans.com/
In my experience, the best corned beef in Cleveland is from the Vienna Meat store. It's a wholesale meat distributor, but they sell to the public and that includes their corned beef. You can buy it warm by the pound with a quarter loaf of bread and a bottle of mustard and have a perfect sandwich in the comfort of your car. It may be a branch of a Chicago company, but some of the guys have been working there forever. It feels like a part of Cleveland to me. It wouldn't spring to mind as a DDD type spot, but the subject was raised in some of the other responses.
Lion on Lorain is a perfect DDD spot. I had a scallop po' boy special on my last visit and it was delicious. They could have served the sandwich with only the cauliflower condiment and it would have been an excellent vegetarian option. I also really like the soup special. On other visits, I had some delicious chili, a delicious creme brulee special, some good tacos. They cook from scratch, they have lots of interesting specials, and the food tastes good. They serve affordable comfort food in a casual atmosphere. That's what I look for when I'm thinking about good DDD spots.
Assad's is just across the street and another good DDD candidate. It's a Lebanese market and I've never seen anyone eat at the two lonely tables they have in the window, but I've eaten there several times. The felafel has been my favorite, but the soudjouk (soujouk? sujouk?) is also delicious.
Kabab G is another Lebanese spot in the same neighborhood. I've had several good meals there, but the best thing I've had there is the beef kababs. In fact, they're easily the best kababs I've had anywhere. They're cooked over charcoal.
Istanbul Grill is a Turkish restaurant in a different neighborhood. They make their own doner (think gyro/shawarma) and it's delicious.
Brunch at the Parkview on Sunday. Weekday breakfast at Shay's on St. Clair. Sokolowski's for Friday night perch and pierogies. WSM for lunch and a beer at Great Lakes across the street. Weekday Happy Hour at Lolita - get the burger and get there at 5:01. Dive bar drinks - Hoople's, Hotz's, Chuck's. Lunch at Marie's (on St. Clair) for chicken and dumplings.
I don't know if you have arrived in CTown yet, but it really has become a great place to eat, both for fancy and casual meals. If you are downtown, head over to the West 25th Street area, near the West Side Market. The Market is a fun and unique place to explore and there is a great Greek Gyro place in it. Just stand in the long line, get the regular and not the large unless you have a huge appetite, and enjoy. Cash only.
Along West 25th street, you can find Phnom Penh, a unique and great, cheap Cambodian restaurant. In case you have never had Cambodian, it is somewhat like Thai, but different in wonderful ways. On the same street, all within walking distance, you can find a Lebanese deli, a Turkish restaurant, a tasty Cajun-low country (SOHO), a byob delicious Mexican restaurant (Orale), The Great Lakes Brewing Company, which makes dozens of their own beers and has good pub grub, as well as many other bars and restaurants. Many of these even have Happy Hour deals, which are awsome.
I am not a big Polish food guy, and many have suggested Sokalowski's, which is food native to many Clevelanders from Eastern Europe, but many people like it and it is not expensive.
If you have a car and want to try something really good, filling and cheap, try Melt Bar and Grilled, which now has 3 locations. You can get a grilled cheese sandwhich with almost any type of filling you could imagine, for about ten bucks. It may be a heat attack on a plate and the places are almost impossible to eat at without an hour or more wait (unless you go at 11am or mid afternoon) but it is great and you won't have to eat again for about a week!
For breakfast, Your's Truly has several locations or there are a few delis, Corky's or Jacks, mostly on the east side of town