There should really be a cross-coutry road trip master thread...
1. Anywhere specific on the east/west coasts? For now I'll assume NYC-LAX.
2. Is leaving the beginning of the week necessary? Most of my trips I left towards the end of the week and spent shabbos in the midwest (usually MKE for me personally) then spent the next week getting to CA. (going West-East would be the opposite. Leave beginning of the week, spend Shabbos in the midwest then go home.)
The one time I did it all in one week I left NY Motzei Shabbos and didn't stop until late Sun night in SD, then spent the rest of the week getting to CA. My cheshbon was that anything between NY and ORD that I cared for I'd already seen. From ORD until western SD is empty farmland that I didn't want to waste time on.
IIRC it's something like 48 hours driving time from NYC to LAX if you drive direct. If you leave motzei shabbos and arrive Fri, that means on average 8 hours of driving per night/day. that doesn't leave much time for sightseeing.
3. IMO you want to get through the eastern half of the country as quickly as possible and spend your time out west.
(I'm working with the assumption that the primary goal/attraction is National Parks etc and smaller off-the-beaten-track type attractions. If you're looking for fancy hotels in big cities I can't really help you.)
The exception to this is if you take a southern route and go through AL, MS, TX etc but you might not have enough time for that.
4. I think what you need to do is figure out the main things you want to see and work around that. Of course if there's a specific city you want to end up in that'll effect it as well.
For example if you want to go to Mt Rushmore and Yellowstone you'll probably want to end in SFO or PDX or SEA. If you want to go Bryce Canyon and Zion you probably want to end in LAX. If you need to end in SAN you won't be going to Glacier National Park.
(of course this is all assuming a ~week time frame)
5. You might want to consider an RV. For example maybe
Cruise America's rolling out of AZ special from PHX to VA. Of course one downside to this is that they're only available at most a few weeks in advance and if you have specific dates you need they won't necessarily have what you want available just then. IIRC
@Galitzyaner did this a couple of years ago, although he did something more local, maybe PHX-SLC?
OK, I think this should be enough to get you started