X2 on Clingman's Dome and Cades Cove - - 2 of the most popular areas of the park (stop at Newfound Gap on the way to Clingman's Dome). Make sure that you stop at the Sugarlands Visitor Center just outside of Gatlinburg to get a map of the area and a schedule of Ranger activities.
Laurel Falls is one of the most impressive in the park and fairly easy to get to. The trail to the falls is paved and about a 2 1/2 mile round trip from the parking area on Little River Rd (TN 73). You can stop there on your way to Cades Cove. Metcalf Bottoms picnic area is also on the way to Cades Cove. From there you can drive back to the Little Greenbrier area. You can visit the Little Greenbrier School (an old one room schoolhouse). Check at Sugarlands to see if the docent/guide will be there to tell you about the schoolhouse. (The 1st week in April may be too early in the season for all of the activities to be happening.)
Make sure that you take the Cherokee Orchard/Roaring Fork Motor Nature Trail that takes you through the mountain area just outside Gatlinburg. Several old cabins and other trails (Rainbow Falls and Grotto Falls) are accessible from the Motor Nature Trail. The Motor Nature Trails begins in Gatlinburg at the "Space Needle" - - I believe that's traffic light #8 on Rt 441 (main route through Gatlinburg).
Another more "off the beaten path" area of the park to explore if you have time is the Big Greenbrier area off of Rt. 321 heading towards Cosby. The gravel road follows the river and eventually ends at trail heads to more water falls (long hikes to the falls but even a short hike on these trails is worthwhile).