Rocky Mountain National Park is a good place to start, if you want to keep with your idea to take I-80 all the way across to Laramie WY, and then down into Colorado. September and October in RMNP is such a great time to be there, with the colours changing and the elk bugling.
Inside the park, Moraine Park CG will still be operating under the summer reservation system through Sept 27, after which B Loop becomes first come/first served. Sites there are typically booked solid through the end of the reservation period, so getting a site could be difficult. Aspenglen CG is open through Sept 27 as well, but is also usually booked completely through the end of the season. Glacier Basin closes on Sept 7, so that won't be an option for you. All of these are on the east/Estes Park side of the park. There's also Timber Creek on the west side, it's all first come/first served and is open until Nov 1. If this is something that would interest you, look in to coming in to that area via US 40 from Steamboat Springs as a viable alternative route. US 40 also could be an I-80 alternative for you, from SLC area all the way across northern Colorado to Rocky.
Some of the private campgrounds in Estes Park will still be open in September/early October.
CLICK HERE for RV park reviews for the Estes Park area. Note that mot of them are pricy; Estes Park CG at Mary's Lake and East Portal are run by the Estes Park Recreation and Parks District, and their prices are more reasonable.
Most of the USFS CG in the area start to close in September, starting the 1st or 2nd week. Few will be open beyond Sept 15.
If you decide to go to Rocky, you could then leave via the Peak-to-Peak Highway (CO7/CO72) that runs from Estes Park through Ward and Nederland and on to Blackhawk/Central City. You could then follow the suggestions from 4runnerguy for the central and SW part of the state, with a variety of options to get there. That Arkansas River Valley area from Leadville to Salida is very impressive, with lots to see and do there.
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