Where are you now? And are you willing to fly around or want stuff within reasonable driving distance? Of course, with 4 weeks you could road trip around the whole country, but it is a big place. Help us narrow it down a bit. It's a huge country and it changes quite a bit from region to region.
I'm in New England (Connecticut) and know the area pretty well. Not much to see in my state, but New York City and Boston are both must-sees if you like city life. I don't, so my suggestions will be more nature oriented. The Green and White mountains of Vermont and New Hampshire are beautiful. Mount Washington is the highest peak on the east coast and is a pretty cool hike if you start from Pinkham Notch and hike up Tuckerman Ravine. You can also just drive up the access road or take the cog-train, but what's the fun in that? LOL! The Maine coast is gorgeous. You can drive right along it and see some amazing rocky shoreline with a bunch of old lighthouses along the way. You'll have a good chance of seeing moose or even black bears if you hike around the parks. If you head south, drive right through New Jersey (it's an embarrassing part of the country, and we'd love it you did us the courtesy of ignoring it.:lol: ) Virginia, North Carolina, and South Carolina all have some beautiful beaches and mountain-y areas inland. I don't know much about it beside going to Myrtle Beach as a kid. In fact, I'll leave the south east to someone else. I won't do it justice, except that the bass fishing is top notch.
The west coast is much more dramatic though. The California coast is unbelievably beautiful, and has the best weather of maybe anywhere on the planet. It's just always 85 and Sunny. LA has some cool things to see, but San Diego is where you want to go for beaches and beautiful people watching. I've been but was there for work so other's will have better info on the really good spots. Personally, I love the pacific northwest. Northern Cali, and Oregon in particular are just amazing. Definitely spend some time around Lake Tahoe. The water is the deepest, prettiest blue you will ever see, and it's surrounded by huge mountains with resorts running gondolas for really good views. South Lake Tahoe (the town) is very cool. Lots of great places to eat, nice beaches on the lake, and great places to hike and camp. South of the lake, there are natural hot springs. So in the winter you can sit in them like a hot tub, surrounded by tons of snow. In Oregon, Portland is the big city to see. I haven't been but I've heard great things. The town of Bend is further inland, but is very cool as well.
If you really like hiking, views, nature, etc. Just tour the big national parks like Yellowstone and Yosemite. You could spend a lifetime in them and not see it all.
Of course, you should swing by Las Vegas for a few nights. Party your ass off and see some weird stuff. Do some gambling, maybe visit the Bunny Ranch, and drink way too much. It's a blast. Just try not to end up married to a stranger or losing your life savings.
There's a lot more to see, of course, and others will chime in, but you have a great opportunity. I'm a bit jealous. I'd love to travel the states for a month. Enjoy it. Don't spend too much time looking through a lens, and tell us how it went when you get back.
