Hi..... I agree about skipping Phoenix.....unless there is something you want to see there (and it will be very hot in July).....go on ahead to Grand Canyon and add a day in LA at least.....
An option that has been useful for friends coming from overseas is to use Las Vegas (LAS) as an airport with low prices to almost everywhere.....and Vegas, baby, is certainly a unique American city.....should be there for at least one night.....many low price hotels available, especially Sun-Thurs nights.
Perhaps, fly DCA or BWI to LAS...... there are many package bus or air trips from Las Vegas to Grand Canyon (or drive yourself, 4-5 hrs)....could go to the Skywalk glass balcony suspended over the edge of the Canyon. (From Vegas, trips go to the North Rim.....Flagstaff area is on South Rim)
Back in Las Vegas, there is likely frequent bus service to Los Angeles, or low-cost flights on many airlines.
For your LA sights, there is likely a sightseeing bus that takes you past many of the things on your list, which would save you driving around.
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For the start of your trip..... coming into JFK, how are you getting over to NJ?
If you are taking trains, with a change to NJ Transit trains at Penn Stn in Manhattan....... Penn Stn puts you in a prime place for some evening sightseeing in NYC, if your friendcares to join you there. A reputable place to check your bags so you can walk around that first evening after the long flight is:
Schwartz Travel Service 212.290.2626
355 W 36th St. (Between 8th & 9th Ave. on 2nd floor)
Close to Penn Station & Port Authority
8am - 11pm
Their website:
http://schwartztravel.com/
provides excellent walking directions (see bottom of their webpage)..... which exit from Penn Station to use to minimze sidewalk time, etc.
As you walk from 8th Av along 36th St, it's about three-fourths of the way to 9th Av...... pass the Wyndham Hotel....and look for sign above sidewalk and on glass door..... enter a narrow hallway to an elevator....go to 2nd floor and the doorway-desk is in friont of you. Allow extra time for a slow elevator and consideraion for non-English-speaking travelers.
......Seems to be a lot of chatter about this place on yelp.com, so perhaps check there for recent experiences (good or bad)......it is basically an extra room of an office.....no special security or receipts, but I felt confident because there's just one person at the doorway-desk of the baggage room, so you talk to the person who is storing it, and see where they put your bags in this little room.....to second what another poster said---do chat-up the person at the doorway-desk, make a personal contact and tip well......lots of international travelers using this place with huge suitcases.....prices for storage run $8 to $15 per bag, but it's fairly informal.....my regular size 22" rollaboard+briefcase was "oh, let's say $15 for both" and I tipped $5.
enjoy!