Hi!
I recommend that you check out Cedar Point in Sandusky Ohio. It's a pretty large amusement park. It also has a water park, but that will be closed for the season during your visit. http://www.cedarpoint.com/
Unfortunately there isn't a whole lot to visit within driving distance of Ohio that involves the interests you mention. For shopping you can visit Pittsburgh, PA. Pittsburgh is by no means a shopping mecca. But the state of PA does not charge sales tax on essentials like clothing. So if you plan on doing a lot of shopping, you can save some bucks. Ross Park Mall is a pretty large shopping center with some high end stores as well as more economical ones. http://www.simon.com/mall/default.aspx?ID=158
Also in Pittsburgh is Kennywood Park. This is a small amusement park. And though it is small, it has some old wooden rollercoasters that are always rated among the worlds best. This park has a long history and has managed to hold onto it's charm. It feels quite different frm the larger, newer parks. http://www.kennywood.com/ During the months of September and October, the park is open only on the weekends.
If you are willing and able to drive further, you can visit the Mall of America in Bloomington Minnesota. I beleive it is still the largest shopping mall in the usa. http://www.mallofamerica.com/ Bloomington is in the suburbs of the twin cities, Minneapolis and St. Paul. The drive from Ohio will take about 2 days.
Another drive destination can be Niagara Falls NY. www.usatourist.com/english/places/newyork/niagara.html. This is purely a scenic attraction. But it is a fun visit. Drive time will depend upon where in Ohio you are staying. But in any case the drive can be done within one day. Be sure to have your papers if you want to cross the border into Canada.
Other than that, I would suggest possibly flying to one location. You mentioned Memphis. That is definitely a great place to visit for music fans. Also, since it is in the south, you'll encounter some warmer weather. You can also drive to Memphis in 2 days. Another great city for young music fans to visit is Austin Texas.
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