How many days? Do you want to take a southern route or the route to Salt Lake City and then south on I-15 to LA?
If you did not want to put all those miles on your car you could take Amtrak from PA or Chicago to LA and rent a car there.
I hope you have a good to great gas mileage vehicle if you do decide to drive. Gas is not cheap in the LA basin.

I'm doing this - coast to coast and back - leaving end of April. But I'm going by train, Amtrak via Chicago and Denver going east and then up the west coast, and Via Rail across Canada coming back.

But lots of people do cross the country by road, and fall is a good time to do it. However, the cost of gas is a consideration. Check your local library for books by people who have done it. The New York Times recently had a blog by someone cycling one way....

Thanks Tom and Thursdays! We were thinking 10 days today if that is even possible. We aren't sure about the route or anything really. It was just something we started talking about. I do like Amtrack idea if they take dogs??? The gas thing is definitely a consideration. Do you know about how much it is in the L.A. Basin? Thanks again for the replies and ideas!

For gas prices see: http://www.gasbuddy.com/GB_Price_List.aspx?cntry=USA

Amtrak - no pets, service animals only - see: http://www.amtrak.com/servlet/ContentServer?c=AM_Content_C&pagename=am%2FLayout&cid=1248543529571

If you mean ten days there and back it won't give you time to stop and see much of anything. Google maps will give you driving times.

<<<We were thinking 10 days today if that is even possible>>>

Possible... yes. Fun... not likely. Part of the fun of a road trip is getting to stop, detour and see what you want along the way. Guessing from Google, it's about 42 hours of driving each way. 10 hours a day in the car puts that at 4 days each way, with just the driving.

It might be interesting to check the prices on flights out to LA and renting a car to drive home. Fly out a day or two before the wedding, enjoy the festivities, rent a car and drive home with plenty of time to detour along the way.

Amtrak both ways is probably cheaper, but that doesn't give you the flexibility of a car.

No dogs on Amtrak unless one of you is blind and need a service dog. The only other dogs I have seen on Amtrak are drug and explosive sniffing dogs.
You could take Amtrak as far as Albuquerque and then rent a car to see a large part of the South-west US. Return the car where you rented (ABQ?) and take the Southwest Chief back to Chicago.
The advantage I see to Amtrak is that you are moving toward your destination all night and you are not paying for a motel room. If you eat in the dining car, you are moving without hitting drive-thrus. Check out the 15 day USA Rail Passon amtrak.com I have used several Rail Passes over the past 10 years.

The thing about Amtrak, though, is that if you want a sleeper they're expensive. (Not as bad for a couple as for a single, though.) Plus, you need to book early - the fares go up as seats are sold. See seat61.com for more info on Amtrak.