There is only one road through the park. The main, prominent, dunes are accessible right next to the visitor center and depending on the weather (which can be very cold in winters) you can hike to the top of the dunes. About a couple of hours will be more than enough to get and hike to the main dunes. Wellington ditch trail is a great place to take sunset photos and trail starts on the opposite side of the visitor center and you can see the dunes from a height.
My son and i stayed fairly nearby and drove an SUV into the park area early enough to catch the sunset for my photography. There are trails for walking and driving if you have the time. We spent several hours even after getting off the dunes. The walk to the top of the sand is challenging but very worth it. There are different levels but once we were in our way up, we couldn't stop. My primary purpose was photography and i got my fill. I love night photos and it's a great place for that. You can spend an hour on the dunes and leave, or many hours enjoying the whole park. Bring a lunch/dinner. The drive up into the hills were a blast. It's SUV time! 😉
You could make it a day trip from Denver (I have done that myself), but I agree that you should plan to spend several hours at the park. You can take a guided tour of the Bison herds, hike or board the sand dunes, play in the water (when available), go off-roading, or take any number of hikes. A full day at the park is best to get the full experience.
The best option is to stay in Alamosa. Realistically you could see a decent amount in a day, but it is an amazing place. In the spring, when the river is flowing you can also swim, but it's fun to bring a sled whenever you go.
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