Liz - the Science Center (right by the Rock Hall) is pretty sweet. There's an AMAZING grilled cheese restaurant in Lakewood (not far from downtown) called Melt Bar and Grilled - one of my absolute favorites! XOXO is a delicious steak restaurant - a little pricey but very much worth it. That is located in the Warehouse District, and there are several clubs/bars in that area. Drinking isn't my thing, so I can't tell you which are better than others, but it is a popular, "hip" place for nightlife.
Our very own Scott Weiland was born and raised in Chagrin Falls - there are some locally-famous waterfalls there (hence the name) and some shopping, if the heat and humidity aren't horrendous.
Tower City (downtown) offers some shopping, though its not great. Beachwood Place is our nicest mall, by far, and is roughly 30 mins or so from downtown - if you're looking to spend some cash.
Hope you enjoy the show and your stay in our little big city.