I'll exchange info on the sweet spots to camp via PM. A lot depends on what type of vehicle you expect to have. The default campground near Red Rock is OK for a place to crash, but has no charm and has lots of folks staging for climbs (which can be fun, as they socialize). Some of the more out-of-the-way places require an overnight permit (they used to be easily obtainable from the RRCNCA site, via listed phone #).
The classic is Bridge Mt via Willow Springs, and the North Peak Ridge. The route can be shortened by 3.5 miles RT or more if you have HC vehicle. You can approach Bridge by a lot of routes right off pavement (Fern Creek), but those are tougher routes for navigation -- easy to get into technical territory. What you wish to do is partly limited by your conditioning; trips off the scenic loop have the requirement that you end your hike/scramble back at the car before the loop closes at 5PM (currently opens at 6AM), unless you get the overnight permit.
I have done lots of scramble routes out there. Some are short but still a lot of fun, e.g.http://hwstock.org/burlap/
This route is so well cairned that it is hard to get lost. The road is a bit rough but suitable for normal passenger cars with decent clearance (don't take your Lamborghini).
The options are so broad, it is hard to recommend just one.
This site lists some trip reports for the Charleston + Red Rock area-- not up-to-date, not exhaustive, but it will give you some ideas:http://hwstock.org/map/01Spring.htm