What kinds of problems?
Rock on! That’s the spirit! As long as you are safe, and having fun you’ll be fine. If you are doing some car camping and day hikes and if there is a ranger station close by always check in with them. And never, EVER car camp on the side of the road. Always do it in an area with other campers. Door locks only keep the honest people out!
Last I heard fire had closed the East Entrance - you can still get in, but it is a long way around vie Red Lodge Montana. There is also talk about restricting access due to rock slide danger generally. Given that you aren’t going for a couple of weeks, you are likely to get in, but I’d call ahead. Go through the East Enterence if you can - Cody is a lot cooler than Jackson.
Having the East Gate closed doesn’t necessarily mean having to go through Jackson - you can come up from the SE on 287 over Togwotee Pass and then hang a right at, well, the Tetons to get to Yellowstone. Me, I like Grand Teton as much as I like Yellowstone, but Yellowstone has great elk in mid-September. The Firehole River area toward the west side of the park is a great place to see them, especially early in the AM.
If you want to see the Wind River Range on the way, though, 287 is the “ugly” side compared to coming up through Pinedale and, yes, Jackson.
I second the Snowy Range Pass / Lake Marie / Medicine Bow Peak recommendation. I used to live in Centennial and still love that area. You can come up from RMNP to Laramie and head west, or from the Granby side head up through North Park (Walden area) to get up there. THen Saratoga, then head NW for Yellowstone…
I recently spend two weeks in WY. We did three two-night backpacks (2 in the Winds, on in the Tetons) and one daytrip drive through Yellowstone as our rest day. Even with limited stops, driving takes up a lot of your time in the big square states. Places like Jackson allow you to hit many things without spending 8 hour days in the car. Plan those days carefully.
There are lots of amazing dayhikes in the Winds that will get you a long way from the road and back without putting yourself out of the way too much. One excellent hike (about 10 miles R/T, not much elevation gain, maybe 1000’) is to the aptly named Photographer’s Point. It’s just north of Pinedale, on the only major trailhead that has a paved road to it, a big plus around there. I forget the trail head name, I have it at home.
http://www.hikethewhites.com/wyoming/titcomb/index.html
The Big Sandy trailhead (south of Pinedale) also offers some great day hikes. It’s a bit more of a drive off the beaten path, which is both good and bad.
http://www.hikethewhites.com/wyoming/bigsandy/index.html
There are a dozen great dayhikes in Grand Teton National Park, mostly trips up deep canyons. Death Canyon is a fine choice, or Paintbrush Canyon. You can basically go up as far as you want before turning around, loops are a little difficult to do as dayhikes.
Yellowstone is much more suited for a day of driving, stopping, and sightseeing. The loop road with planned stops for geysers, hot springs, and animals, plus the Yellowstone river waterfalls. We stayed in Jackson for two days after our Tetons backpack and it made for a long day. Staying in Yellowstone would make for an easier day.
But anyplace you decide to visit will have great things to see. I still think the key is limiting your driving. Many places are nearly a full day apart once you get on the smaller roads, don’t spend the majority of your trip in the car.
I had my first Colorado experience in June. We did Denver and Colorado Springs for a family reunion.
We did all the CO stuff you have - we stayed right by Garden of the Gods and went there a couple times, Pikes Peak Cog train, Royal Gorge etc. But one of the neatest things we visited was the North Cheyenne Canyon Park. My uncle was out trying to find some store and got lost and ended up there. We went there a few times on our trip - it seemed to be a place that locals visited more than tourists, and it was very beautiful. Fun hiking (probably not ‘serious’ hiking at all though) and driving trails, waterfalls. Not crammed with tourists. Most people were there walking dogs and they were all really nice too - one lady said she just loved my green toenail polish.
Here is one of the waterfalls there.
::waves from Grand Junction::
You might want to take just a slight detour to Colorado National Monument. You can drive through in an afternoon, or stop and take some of the trails. The visitor’s center is very good.
The Rockslide in downtown GJ makes good beer!
And it’s cheap too - a coupla bucks for their sampler of six or so beers. I recommend the porter and the pale ale.
Thanks again everyone for the contributing to this thread. It’s all great and useful advice.
I’ve made reservations for two nights at an inn in Estes Park, so it looks like I’ll be driving to Colorado Springs, going up Pikes Peak in the morning and then driving to Estes Park where I’ll spend a couple of days hiking in RMNP. (Pikes Peak is just something I gotta do. When I was there a couple years ago it was October and I couldn’t get quite to the top because the road was closed.)
Does anyone have any recommendations for specific campgrounds in Grand Tetons? I figure by that time I’ll be more than ready to sleep on the ground and be woken by birds. 
Also, I’ve never car camped in bear country. Are there bear lockers or anything like that around?
Yes there are bear lockers - sort of - they recommend sealing all food and waste and that is usually done in a sealed dumpster or your own container. I’m nt sure about actual rentable bear lockers.
Also, this time of year bears will be gorging to prepare for the long winters…But when car camping, I’d be least concerned about bears, and more concerned about people and general riff raff. You are more likely to get mugged than attacked by a bear whilst car camping. Not meaning to scare you, just camp near other folks… I was talking to a national park service ranger and he said person to person theft and conflicts have been up in the past 5 years…
In RMNP, there were bears about (one car in Moraine Prk CG had been broken into the night before I checked in - the guy had food stored in it and left his car window cracked), but they say storing food in cars is safe. There are not enough bear boxes to serve everyone in the campgrounds. After the break in they wanted coolers and food boxes stored in cars hidden from view/covered up, too.
I didn’t see anyone hanging their food; most people just put it in their cars out of sight. They will ticket you if you are away from camp and leave anything out. - and I did see rangers going though people’s camps one day when I was back in camp at lunchtime. I’d just keep everything but my tent/pad/sleeping bag/sleeping clothes stored in my car and get stuff out/put it back as needed. I did not cook other than to make a pot of coffee on my camp stove in the AM.
I’m north of Colorado Springs in a charming squared bedroom community now, but…
I grew up in the even more quaint (and surprisingly moneyed) Cheyenne Canon. In the picture, that’s Helen Hunt Falls (named for the novelist Helen Hunt Jackson) at the far west end of North Cheyenne Canon. At the far west end of South Cheyenne Canon is Seven Falls. It’s privately owned and an admission fee applies, but it is worth it. Both are worth a look and the hiking, wandering, picnicking options in the canon are worthwhile. The Starsmore interpretive center between the entrances to the two canons can provide information about seasonal goings-on and predator activity (mountain lions being more prevalent inthat area than bears, though you are more likely to see the bears). I lost a few high school friends in hiking accidents and am familiar enough with the local Search and Rescue activities to suggest that Turek not go all the way to the top of the Mt. Cutler trail alone and that, if he is not a rock climber, he not start thinking he is one in the Cheyenne Canons with their crumbly sandstone and Pikes Peak granite.
To get to the canons (they use the Spanish spelling and I can’t find a tilde for the “n”), take the Tejon Street exit from I-25 at the south end of Colorado Springs. You’ll pass the Blue Star restaurant (sorta toney, but tasty) and a microbrewery. Go south briefly and take a right on Cheyenne Boulevard. You’re in a 1920s to 50s era residential area a few blocks north of the Broadmoor Hotel. Stay on Cheyenne Blvd. You’ll pass 8th Street and Cresta Road (among others) and eventually arrive at the Starmore Center, which is an old stone house that was moved into the canon at great expense and incovenience to save it from being demolished.
If you’re looking for a place to eat in Manitou Springs around your cog train travel, consider Adam’s Mountain Cafe which has recently moved and expanded into the first floor of the old Manitou Spa building, next to the Penny Arcade (which you’ll know when you see it).
Aaaand, don’t trust bears to respect your car if that’s where you choose to hide your food. I once watched, through binoculars, from quite a long ways away, as a bear peeled the hatchback off a Datsun to get what looked like a bag of marshmallows. It’s kinda a cost benefit analysis situation. (For the camper. And it was only a Datsun, but I bet it had a higher book value than the marshmallows.)
Jenny Lake is IMHO the most desirable (great setting, tents only, etc.) and therefore the hardest to get into.NPS Web Site
Do you like trains? If so a visit to Golden Spike is a must, it’s where the rails from the Pacific coast and east met. It’s just north-west of Salt Lake City. You get to ride in the steam trains too.
What would be your second choice? Getting there before 8am will be problematic.
Not so much. I’m more into nature things and ruined things.
I’m also a fan of waterfalls. What are some good waterfalls to see?
Due to circumstances beyond my control, it looks like my trip will cut from two weeks to one.
OK, pick 1 or 2 places to concentrate on and spend most of your time there. The area by Jackson Hole/Tetons/Yellowstone is enough to keep you enthralled for a week without seeing the same sort of thing over and over again, and without spending 40% of your waking hours behind the wheel of a car.
Yeah, I’m thinking of focusing on RMNP and Grand Tetons. But man, it does seem like a long way to drive for just a week. 
Damn work! Even though I asked off a couple months ago and was assured it wouldn’t be a problem, a lot of work has come in. So my boss asked if I’d not take off and the company would compensate me rather than having to hire a temp person. She assured me that if I DID want to go anyway, I could and she wouldn’t have a problem with it - she recognizes people’s need to get away. As a compromise, I’m going to offer to only take a week off and take half of the compensation she offered, which she was amenable to.
You didn’t mention it in your OP…how about a run up to Capitol Reef? You could go up via 261 and see the Goosenecks, drive up the Dugway, catch the view from Muley Point, check out Natural Bridges. Take the ferry at Halls Crossing, Then up to Hanksville and over to the park. If you’re adventurous, and have a suitable vehicle, you could take the Burr Trail up to Boulder and then back around to Capitol Reef.
Define “suitable.” I have a Toyota Tacoma. High clearance but 2WD.
More detail, please. Is that in Colorado? The only name I recognized was Boulder.