Try going to Ohio City, Tremont or the Downtown districts of Playhouse Square, Gateway, Warehouse district or the Flats. These areas are easily accessible by the Red Line Rapid (subway)
The St. Clair - Super area is full of lots of artist lofts and Asian Village with some unknown but AWESOME asian restaurants, but you also find ethopian, somilian, Dominican, Puerto Rican restaurants and this area is home to the worlds best corned beef restaurant.
University Circle's cultural institutions are second to none, although the CMA is closed for a massive expansion, there is still plenty to see.
Buckeye-Shaker Square/Larchmere, is a old school neighborhood sorta like NYCs upperwest side or Brooklyn Hts. Diverse, great restaurants and stores, accessible by the Green/Blue Light Rail from Tower City.
Edgewater is nice, some nice restaurant and nice views of downtown and the lake.