So you do that Yellowstone bus tour I told you about, the train tour MmeLeota told you about, some days at DLR, and you're up to about 12-13 days including travel time. So there are a couple things you could do to round our your two weeks:
1. Tack on an extra day in California to go to the beach. Pacific Ocean!
2. Add a day or two to Yellowstone. Seriously, there isn't enough time to see it all.
You can do some backcountry camping if you're really brave (bears! bugs!)
3. Spend a little time in Sonoma, AZ, which is really pretty
One thing to keep in mind, there ARE commercial flights available from Phoenix to the Grand Canyon. They have a little information on the park website.
http://www.nps.gov/grca/index.htm And it would be nice to spend a day in Old Scottsdale, but keep in mind that June is the "slow season" for Phoenix, and it's REALLY REALLY hot there. Don't believe any of that nonsense about it being a dry heat. Hot is hot. It frequently gets over 100 degrees Fahrenheit, which is obnoxious. So, many of the touristy kinds of places are closed at that time of year. But you'll be there at the tail end of "spring" so the desert might yet be green(ish) and not all brown and dried out. April is a BEAUTIFUL time to be in the desert in Arizona. I haven't been in June. And again, you'll have to drive to see it.
Now, if you want to extend into the third week of your trip, I recommend one of the following:
1. Rocky Mountain National Park. It's reasonably near Denver, and I imagine that if you are able to get transportation to Grand Canyon and Yellowstone, you should be able to arrange something similar to Rocky Mountain.
2. Glacier National Park in Montana. I only mention this one because the glaciers are rapidly disappearing and within the next 25-30 years they might be gone. It's much more remote than even Yellowstone is, though, and you might be forced to rent a car to get there. I do know there is an Amtrak train that goes to the region, but I don't know anything about it.
3. Spend some time in Chicago. GREAT museums and food, good public transportation system. You'd be in town during baseball season, and a Cubs game is always a good time if you can stand sitting in a hard seat for 3 hours+, eating hot dogs and drinking bad beer.
4. St. Louis Gateway Arch. I think you might find this interesting, and the museum beneath is pretty cool. Gives you some American history.
Not more than a day, though. It's pretty easy to get there from the airport via public transportation or taxi.
5. Mt. Rushmore in South Dakota. Again, it's kind of in the middle of nowhere as most of our parks and monuments are. But it's in the midst of the Badlands, which is a really rugged, yet beautiful, part of the country. Not really mountains, and not flat plains.
If you really don't want to rent a car you're going to be limited. We Americans love our cars and our public transportation system as a whole really stinks. There are exceptions in some areas, but they're few and far between. We all just drive places all the time.
One of the things you might consider is visiting the major cities or areas where us Subsonic folks are living. Might not be the National Parks but would definitely be interesting. You'd see the "real" American countryside. Just a thought. You'd be welcome here in Indy. I'll take you to the track and the Indianapolis Motor Speedway museum.
The race is always Memorial Day weekend but the museum is open year-round.
So there's some more stuff to think about. Of course, we haven't even considered anything in northern California (Sequoia National Park) or the northern Pacific Coast, nor have we talked about the lesser-known but equally beautiful parks in Utah, Colorado, and Arizona (Zion, Arches, Bryce Canyon, Mesa Verde, Petrified Forest). And now you know why people have visiting all of the National Parks on their "bucket lists".
It's just too much to do even in one year.