For someone *new* to backpacking, I too would recommend the North Lake - South Lake loop, which is the same as the Piute Pass - Bishop Pass loop. Here's why:
* Some of the most spectacular scenery you can reach *on-trail* in the Sierras
* Choice. Once you're on the loop, if you're feeling humble or altitude-affected, you can just stick to the trail. On other days, you might feel more ambitious and willing to explore side canyons or off-trail. From north to south, the interesting off-trail places that come to mind are: Humphreys Basin, Darwin Canyon, McGee Canyon, Davis Lakes, Ionian Basin, Ladder Lake, Dusy Basin.
* Perfectly ideal for 9 days (for the reason given above).
* The East side! Your trail starts and ends on the eastern side of the Sierras, which is simply more spectacular than the west side. There is about 6000-8000 feet of dramatic relief from the 13-14k mountains down to the desert below. The Highway 395 through this area is beautiful to drive just by itself. If you start in the Bay Area or Sacramento, you access the eastside/395 by driving the Tioga Road through Yosemite, also a huge pleasure. Once on the East side you can check out Mono Lake, the Mobil Station/Whoa Nelly Deli in Lee Vining, stuff in the town of Mammoth, natural hot springs outside of Mammoth, and places like Schatt's Bakery in Bishop. There is nothing remotely equivalent to this experience on the Western side of the Sierra.
Get your permit well in advance! This is an extremely popular loop.
The trailheads at North Lake and South Lake are a few miles apart. We simply parked our car at one trailhead and hitched to the other. When we ended our trip, our car was there waiting at the exit trailhead. No need for complicated shuttle planning.
The Rae Lakes loop would be a remote second choice. That loop is not as diverse, it takes a bit longer to achieve the truly majestic backcountry, and it's a West side entry so you would not get to enjoy the Eastside landscape or the Yosemite drive-through. (Alternately, you could do a non-traditional, IMO prettier Rae Lakes loop from the *east* side, starting at Kearsarge Pass and ending at Baxter Pass. However, I would not recommend Baxter Pass for beginners! - and the shuttle is more complicated.)