We would like to take a trip with our 3 children ages 16,18,21. Any one have a suggestion on a place to go that everyone will enjoy?

Try Disney World FL (I first went when I was 23 and LOVED it) including Epcot and Pleasure Island, also Universal Studios FL, or Southern California, incl. Hollywood.

Take them to Europe either Paris or Rome will be cheaper than vacation in WDW

Sue:
What have you already done?
After we did the Disney vacation twice, we moved on to the Hard Rock Hotel (Orlando) & Universal Studios. If your kids have not done Universal, then that would be great fun if you are all into the amusement park theme stuff.
If you've already done Florida theme parks, what about going "out west" to our National Parks?
Finally, you could always check out the Caribbean. There you all could snorkel, or learn how to dive together.
Just depends on what you and your family enjoy and want to experience together. (Also depends on your budget).
There's lots to do right here in the USA. Yes, it would probably be "cheaper" to take them to Europe than a WDW vacation. But, I personally want my children to see as much of our country before I take them overseas. And I say this knowing I will be slammed. But, I do love Florida. Just my opinion.
p.s. Talk with your kids!! They may have some great suggestions.

YELLOWSTONE

We've done many of the National Parks and WDW in the past. Loved them all. Last year I did Europe with my sons (20 and 18). It was incredible. My older son was studying abroad in Bonn. I met up with him. We traveled by train for a week - Munich, Prague, Zurich, Interlaken (known for extreme sports which your kids would love!) - before meeting my younger son in Paris for 3 days. Then on to London for 3 as well. For the first time in my life I packed in a carry-on. I even showered in a McShower in the train station in Munich. Bought everything from Chicos and Travel Smith. Washed things out a night as needed. Packing light made the trip so much more enjoyable. Loved our hotel in Paris. Little, boutiquish, and cheap! I would go again in a heartbeat!

They are a great age for Europe if they haven't been. If it's for a week, a would concentrate on a major city (paris/london/rome), if you have more time, you can branch out more. Much depends on you children's interests and the time of year. My kids are younger but just loved Italy (and so do I). In US, besides what has already been mentioned (although I would choose most anything above WDW), California coast (SF, Big Sur, LA, etc) is great.