I'd second the suggestion to visit downtown Greenville for either 1/2 day or evening. If you enjoy a good burger, Grill Marks on Main Street is a good spot for lunch (great shakes too!) Check out the Mast General Store (on Main St across from Hyatt hotel) and stroll down Main St to Falls Park and Liberty Bridge. Also, note the Clemson B-school building at the intersection of Coffee and Main - they hold events open to the public in that building, so you may be able to take the elevator up and check out the space - it's new and has nice views.
You may also want to stop in Easley on the drive from Clemson to Greenville. I think some people that work at Clemson choose to live there as it is located between Greenville and the University.
If you have kids with you and the weather is nice you might want to check out the schedule for the Greenville Drive. It's a fantastic venue and tickets are very reasonable.
Fluor Field at the WestEnd (Greenville, SC): Hours, Address, Attraction Reviews - TripAdvisor
I'm not very familiar with hotels in the Clemson area, so I'm afraid I can't help you there. But I usually have good luck with using TripAdvisor recommendations so you might want to check that site.
Have a good trip and let us know what you think once you visit!