Stay downtown if you want to party
Anything near or in the Arena District is a good bet. However, do not be affraid to leave the downtown area, as the areas just barely outside of downtown are the most interesting and happening sections of the city.
In downtown the arena district is always happening thru-saturday. Then you should take a cab up high st. into the Short North and check out all of the local small bars/restaurants. Then if you dare travel a little further up high st. and enter all off the craziness that is the Ohio State University. All of this is within a mile of each other and would be best to travel between by cab.
Some hotels in the Arena District include:
Red Roof Inn
Hampton Inn (highly recommended, this is a very very nice hampton)
Crown Plaza (has wireless and is comfortable but some renovations may be needed)
Hyatt Regency
Lofts Hotel (good if you want a slightly expensive but nice boutique hotel)
Drury Suites (nice yet family oriented and less expensive)
All of the above hotels are located on Nationwide Blvd. Right in the Arena District near a lot of bars/restraunts and within a cab drive to the OSU campus and the Short North which are also very hot for nightlife and boutiques.
If you do want to see some class A shopping then you would need to leave downtown and take a trip to Easton Town Center (Nordstrom, Puma, etc..) If you want to go to a SAKS Fifth Avenue then a trip to Polaris on the north side is a must. Other than these shopping trips you will be able to find everything you want by staying in a downtown/arena district hotel.