Check here for Ellis Island and Statue of Liberty:
http://www.doi.gov/news/040803b (One can still tour)
Check here for Hollywood:
http://www.trustedtours.com/store/Los-Angeles- C428.aspx
(Closest beach is Santa Monica - the pier has lots going on and the food is good, quite a distance from Hollywood though)
A word about the weather:
The Grand Canyon has a mean elevation of over 5,000 feet above sea level. It'll be cold, windy and blustery and bone numbingly cold that time of year.
The "Great Lakes" is a huge area, from Buffalo, NY on the east, to Chicago and Milwaukee on the west. Same thing about the weather... fair warning.
I would suggest that you do a little planning. Although a month seems a lot of time, remember the U.S. is 2,500 miles distance coast to coast and as much as 1,800 miles from north to south. A lot of it has a lot of vacant space in betwee. I live in the inter-mountain area of western U.S. and it's 100 miles to the nearest larger town, 500 miles to a large city like Denver or Salt Lake City.
Contd.