I would suggest flying into NYC, and of course then go to the Bronx Zoo. You could go south from there to Philadelphia, Maryland Zoo, Baltimore Aquarium, and National Zoo. All of that is in pretty close proximity to each other, and the Maryland Zoo and Balt Aquarium(one of best in country) can be done in the same day. So that can actually be done in 4 days, but this is also the best site seeing area(at least in terms of cities) in the country, so you should spend more time there. You could eliminate everything here other than Bronx if you don't care that much about seeing the cities, because other than the Aquarium, none of the other establishments are great, all good though.
Otherwise as Arizona Docent said, San Diego is a must for it's zoo and animal park. Sea World is also great. Southern California is another of best places in the country to visit with it's great beaches, great weather, scenic drives, big and exciting cities, and tons of touristy things to do.
Unfortunately my plan only brings you to 3 of the top 10 or so zoos in the country and one of the best aquariums, however like I said, if you want to do see some great things other than zoos, this makes it a great combo.
Plus, there could be some more time to visit another area and its zoos. Maybe Florida, Chicago area, or as Black Rhino said, the Midwest offers a lot of very good zoos with Henry Doorly, St. Louis, Oklahoma, Kansas City, and Sedwick Park. I wouldn't classify Minnesota as being in this area, that's why I left it out.