The obvious answer would be to go to Rocky as it is relatively close to Denver and would as you say give you the most "bang for your buck." However, there is a fire burning in the park right now. You may try going south to say the mountains west of Colorado Springs. A lot of campgrounds (at least NFS)are closed for the season and getting a spot in those that are open may be a challenge on a weekend so close to Denver. Depending on the weather as well as what you find available, you may just want to forgo the camping idea (along with hauling all the equipment) for staying in a motel.

If you do enjoy camping and have to take a pass on it this time then definitely go back to Colorado when you have more time as the entire state is a camping paradise.

The fire is still burning, but in a remote area; however, Bear Lake Road and Moraine campground are closed. It's a big park, and plenty of camping is available; check with the park, or http://www.estesparknews.com/.
The wind is really stiff right now; I hope that the fire doesn't reinvigorate.

Camping at high elevation in late October is likely to be cold.
If you can afford it, try the Hotel Colorado in Glenwood Springs. The world's first Teddy Bear was made there. http://hotelcolorado.com/history.html
Next to the hotel is the world's largest hot spring fed swimming pool.
Doc Holliday is buried in the cemetery on the top of the hill.
Aspen is only 41 miles from Glenwood Springs.

As of yesterday's update, Bear Lake Road is reopened. Moraine Park campground is still closed. That leaves just Aspenglen campground open on the east side as Glacier Basin is also closed due to road construction.

For a first time camper, Moraine Park would have been my recommendation as it has the best views and is closer to where you'd want to go hiking. But with it closed and this just being a one-nighter, I'd vote for a cheap motel room instead of taking all that gear.

There are trail restrictions/closures as well, but there are so many trails that you'll still be able to find some great hikes for Saturday/Sunday. As of now Trail Ridge Road is closed but it may reopen by the weekend. If it does, for sure you'd want to drive that road for the views.

Depending on your fitness, I'd hike the trail/route in Upper Glacier Gorge or hike to the Loch and then Sky Pond. But Upper Glacier Gorge would be my first choice. Just get started early, thunderstorms shouldn't be an issue but you'll want to allow plenty of time to stop and rest due to the altitude.

If you do a lot of hiking on Sunday, there are pay showers at Dad's Laundry in Stanley Village (in Estes Park).

I would not recommend camping in October. For very simple accommodations, you could contact the YMCA of the Rockies. There is one in Winter Park, called Snow Mountain Ranch. And one in Rocky Mt Natl Park. They have dorm type rooms, cabins, a lodge, etc.

If you can save Rocky for a time when you have more days, you can camp in the park, and there is not a fire do so. We were last there about a month ago and found the park pretty hazy, and the fires were in Wyoming.

The other issue you have is that it snowed pretty heavy in the mountains yesterday. So many of the trails could be snowy and/or muddy since it will melt at lower elevations. This is not the best time of year for quick camping in the mountains. The weather can turn very quickly and be miserable if not well prepared. It is cold and windy today which means the weather is changing.

Also meant to warn you to allow double amount of time to get back to the airport on Sunday afternoon. The traffic out of the mountains into the city on Sunday afternoon can range from very bad to awful. This weekend should be in the bad range since there is not much going on in the mountains this weekend. But error on the safe side.

Boulder is close - 30 miles or so - and it's a great college town and you could easily go on some hikes using that as a baswe.

Boulder is a great idea, especially if you are young. Or young at heart!