New to the city, friend coming to visit: What should we do?

Definitely check out Conventry area. Tommy's is an awesome little restaurant with great shakes. Also hit up the West Side Market.  If you can, hit up Swensons (7 Hills) Drive In Restaurants....this is more of an Akron establishment but there is one somewhat close to Cleveland. 

I'm from the eastside and over here ;) consider these choices: Chagrin Falls. The shops are local and personal and a good number. You can enjoy the falls as you walk through town. Great choices in food. Lemon Falls is my personal favorite for healthy, local ingredients made fresh and wonderful baked goods - you just can't go wrong there. They have a few tables, it's cozy but closes at 5:00 and not open on Sundays. Yours Truly has been around forever but there's a reason - great food. Burntwood Tavern - beer, great food, atmosphere and fun.  (Service can be slow.) They also have live music but I haven't been there for that. Chinese restaurant (name?) is really fresh and good. For music and more beer, there is also Rick's Cafe. It's been around forever but I haven't been there for music. Every one seems to love it. 2. University Circle/Little Italy (Ohio) - oh yeah! The food! Mama Santa's. Lots of music, beer, galleries and shopping. The museums, of course are there. I used to go to small jazz/music clubs and a fun bar, The Brick Lofts at Historic West Tech High, down there but I haven't lately so not sure of them or of prices. On Euclid, near Mayfield, there used to be some great clubs. The restaurants vary in price around the museums and Case Western. Mama Santas in Little Italy is very reasonable and the pizza there is worth a trip in itself - it's good! Then wander to a bakery down the street - they are to die for too.  OH, and one more spot that I haven't made it to yet, but I will. I have heard great things...(it is a bit of a shhhh... secret... ) The Velvet Tango Room. You need to dress for it. It has history, music and the art of cocktails. Have fun!

Rock and Roll Hall of Fame and Museum is a must-see. Explore Tremont and/or Ohio City: Great Lakes Brewing Company is one of my favorite breweries and Lucky's Cafealways satisfies. Parkview Nite Club is a dive that has great food and music (check schedule). Another great venue for music is The Beachland Ballroom & Tavern. Everybody in Cleveland loves Melt Bar & Grilled (several locations) and McNulty's Bier Markt (the latter can get pricey). If you do feel like splurging, go to The Velvet Tango Room for a gin fizz cocktail  - they are the best in the land.

Ordinarily I would never send someone to a shopping mall for adventure, but the The Arcade is really a lovely place to see. It's a shopping mall from the Victorian era that's been largely preserved--people even have weddings there. Even if you just have coffee there it's worth seeing. 

You can't miss with a visit to The Cleveland Museum of Art. If you want to impress, drop in for a drink Provenance, have a bite at Doug Katz's most excellent "cafeteria" next door, or hop over to Michelson & Morley on the CWRU campus for a delicious lunch or dinner that won't break the bank.  

For cheap eats, you can't beat Aladdin's Eateryjust up the hill in Cleveland Heights. Best hummus and schwarma ever.  

The scene on East 4th Street downtown is lively and filled with all kinds of food and drink.  

Very few places have this:  a visit to the USS Cod, a WWII submarine docked and open for exploration just north of Rock and Roll Hall of Fame and Museum, another spot to explore if you're into music.

Cleveland is brimming with great things to do - while you're humming around University Circle, try Cleveland Museum of Natural History, Cleveland Botanical Garden or to catch some local history:  Western Reserve Historical Society, although only the The Cleveland Museum of Art is free to all, each and every day.  What other city can say that?  

The art museum is an awesome idea! Check out Coventry area for some fun, cute shops like Big Fun. If you're into vintage, look into Flower Child, it's really fun to just walk around and look at everything. I know it's cold out, but Edgewater Park is a great place to see the lake and there's a pier that you can go out on to see a great view downtown. You may also want to look into Gordon Square area for some local shops, or if you like ice cream, try Sweet Moses.