I've been to quite a few, so it depends what you're looking for:
Rocky Mountain, CO: Good backpacking, plenty of wildlife, within a 2-hour drive of Denver. Absolutely beautiful meadows and mountains. Trail Ridge Road might be closed into July, and then it gets pretty busy, especially on weekends. If you were to do this, try to arrive midweek and be in the backcountry from Friday-Sunday.
Mesa Verde, CO: Do it in conjunction with Grand Canyon or the Utah parks. Awesome cultural experience, but not worth a vacation itself.
Utah parks are spectacular, but they're a long way from a decent-size airport. St. George is probably the closest one, and it's an hour from Zion and further from the others. Personally, I'd try to take two weeks and hit Zion-Bryce-Capitol Reef-Arches-Canyonlands in series. There are some major similarities betwen all of them, but they're surprisingly small, and it's hard to string together a week's worth of backcountry in just one. [Been to Zion and Bryce, other three are hearsay from buddies in CO and UT]
Carlsbad Caverns, NM; Guadalupe Mountains, TX: I lump these together because they're 35 miles apart. The best way to get there is to fly into El Paso, and then it's about 2-2.5 hours to GMNP. Guadalupe has the tallest point in Texas, but the rest of the park is pretty amazing, too. Lots of scrubby plants, arid climate (kind of like high desert). There's a couple of sections of the park that are accessible only by hiking in, and these are apparently the place to be. Carlsbad is probably 2 days' worth of entertainment if you go on a couple of the guided tours (which cost a little bit, but not a ton).
Joshua Tree, CA: Amazing rocks for climbing. If you're into climbing at all, I'd recommend checking this one out. About 2 hours from LA. Camping there is dirt cheap, since most of the park is wilderness. Basically it's completely unstructured and you can just go off and be unsupervised for a week. No reliable water sources, so you'd be best served doing 2-3 days max at a time. One awesome feature is that the park is right on the border between the Mojave and Colorado deserts, and the change is obvious. Wet half of the park gets about 4 inches of rain a year. Really, really hot in July and August.
Redwoods NP/SP, CA: ~6 hours from San Francisco, ~5 from Portland, other small airports. Main attraction are the trees, and you'd have to do some hunting to get a week's worth of backpacking (It's definitely possible, though). Fantastic day hikes abound, but you have to drive between trailheads. National and State parks are run jointly, and it's really spread out. Not really lots of mountains, but you may end up getting numb to the trees after several days there.
Crater Lake, OR: Spectacular place, easily one of the coolest things I've ever seen (http://www.naturepicoftheday.com/npods/2008/august/crater_lake_1680x1050.jpg). It's not boring by any stretch after a few days, but the park is pretty much centered on the lake, and hiking is pretty minimal.
Yosemite, CA: Actually, Half-Dome permits become available on March 1, so you might be just in time to sign up if you're lucky. Haven't been yet, but word on the street is that it gets too busy in the main portion to have much fun, but... as soon as you get onto the trails, the crowds drop hugely. Try to scout out parts of the park that aren't road-accessible and you should have no trouble. Everyone I know who's been there has come back saying it's magical.
Sequoia/King's Canyon, CA: Like Yosemite, but less busy, is the rap I've heard. Really active bear country (which is intriguing but not enticing to me). Awesome landscapes. [Hearsay]
Mt. Rainier, WA: Go here go here go here! "The Mountain" is amazing for backpacking and camping. March 1 should be the open date for reserving an itinerary on the Wonderland Trail, which is about 93 miles long and circles the mountain. If you're really accustomed to backpacking quickly and with lots of elevation change (as there are 26 glacial valleys/ridges you have to go up-down-up-down the whole time), you could probably manage it in a week of hiking. Within 1.5 hours of SeaTac.
Olympic, WA: I really have to plug this park, since I worked on the peninsula for a while. It's kind of lost in the NW because its highest point is under 9000 feet, but its claim to fame is the diversity--about 5 different distinct ecosystems. If you were to go here (which I can't recommend highly enough), there's the opportunity to do a couple of days hiking on the coast (literally, on the beach, dependent on tides), then in the span of two days you could start in a temperate rainforest (though the summer season is a much milder climate) and then get into glacial valleys on Mount Olympus. Plenty of trails and about 95% wilderness give you the opportunity to string together a loop and see plenty of different features. Probably 2.5 hours from SeaTac, depending on where you wanted to start hiking. However, you could drive around the peninsula in about 5 hours if you were ambitious and didn't stop (but why would you do that?).
North Cascades, WA: Underrated park. Has something like 30000 visitors a year (by comparison, Rainier and Olympic are both over 2 million and are the same distance from Seattle). Backpacker's paradise, as it's almost totally wilderness. Lots of loop opportunities, valleys, glaciers, well-maintained trails, etc... When I went last year for a 5-day trip, I didn't have to pay a thing for camping and saw a total of 5 people while I was on the trail (in mid-August!). If you went in mid-late September, you might happen upon a few early-finishing PCT through-hikers. Definitely look into this one.
Glacier, MT: Kinda hard to get to without driving a little while, but it's totally worth it. Great backpacking--it's really geared to people who want to get off the roads and onto the trails. Designated sites are all maintained really well, as are the trails. The more I think about it, the more I want to go back.
I can't comment on Badlands or Grand Teton. Post back if anything I've said has intrigued you and want more.