There are 2 ranked dunes (Dune 8860 and High Dune), and 1 named dune (Star Dune). Star is probably the hardest to get to, while 8860 is probably the least climbed because it isn't named, and slightly more obscure to pinpoint. All are worthwhile, especially since you're in the sand already.
Most people climb High Dune:
People dedicated to getting both ranked dunes can take discontinuous ridgelines from High Dune to Dune 8860, then follow Medano Creek back to the original TH.
Medano Creek crossing
Take the Medano Pass road as it goes along the east side of the peak, on the east side of the park. The road for Medano Pass is BEFORE the campgrounds, and is not well marked. Drive north on the road - we had two vehicles (one of which had fairly low clearance), so we parked at the infamous Point of No Return. The road continues and gets closer than we parked (you can park .3 miles closer than we did).
From there, head west down to Medano Creek. You will likely need the GPS coordinates to correctly identify which dune is Dune 8860. Keep thoughts of giant worms out of your head.
There is a park admission fee necessary to enter GSNP all year long.
The heat in the middle of the day in the middle of the summer is also probably prohibitive.
Camping is available in the park.
Other area campgrounds
External LinksGreat Sand Dunes National Park