This looks like about a 6 day drive, ballpark. There are many other route options as well, but if I were pressed for time, I'd probably use this combination: Fastest way is mostly or all interstates. Get over to I-81 and use that and I-40 to get to Nashville, then up to I-64 from Nashville and take it to St Louis, pick up I-70 to Denver from there. From Denver, I-70 west to I-15, then I-15 to San Diego. (My original thought was I-25, I-40 & I-8, but sometimes I forget that not EVERY route west needs to come through AZ! I suspect I-70 and I-15 will be faster in this instance).
Bob