Where are the best places to visit/ which route is best?

THe most direct route is 43 hours of driving (about 3000 miles), which if you have to do it in 7 days is at least 6 hours of driving a day.  If you do the most direct route it would be via I-90 W and I-80 W.  I'm going to suggest that you do more driving in the east so you can visit a few more places in the west.  So Boston to Buffalo (7 hours).  You might want to deviate a bit and go to Niagara Falls if you haven't been there.  The next stop will be in Gary (8 hours), then Omaha on the Mississippi (7 hours).  Then head for Denver (8 hours).  From here you go through Utah, and I'm suggesting that you head for Bryce Canyon National Park.  It will take almost 9 hours to get there, but you can spend the morning there before heading for Las Vegas (4 hours).  And you can spend the morning in Las Vegas because Los Angeles is only another 4 hours.