T2 was the old international terminal before Bradley was built for the 1984 Olympics. Once Bradley opened, they rebuilt T2 and the main airlines were Northwest and Pan Am with a host of international flights. I flew out of there a couple weeks ago on Air Canada and the security lines were long and slow moving (only one security gate was open), so it's not necessarily a hidden gem.

But back to the question. T2 to T7/8 is a short walk across the middle of the terminal area, but if you have a lot of bags, it's a pain. Assuming you have the time, the shuttle will not take too long. It sounds like you're buying all these flights on one ticket, and that's important. If you do it that way, they'll be responsible for getting you on a later flight if your other flight is late. The other option which is slightly riskier is to just buy the flight out of LA and then buy a ticket on Southwest to Tucson which is right next door in T1 at LAX. But if you miss the flight because they're late, the other airline has no obligation to accommodate you.