First off, I'd suggest going to Val d'Isere first and then La Grave.
You're probably going to have jet lag the first few days that you're in Europe because it's a long, overnight flight from the US. Also, if you're not familiar with skiing mostly above tree line, then you'll probably need some time to get used to it, especially the bad visibility on cloudy/snowy days. Val d'Isere will be more forgiving for those first few days, while La Grave is entirely off piste and is best done when you're not jet lagged and you're used to skiing the particular conditions.
Honestly, unless you're particularly good at avoiding jet lag and you're quite familiar with high alpine skiing, I'd suggest spending all 6 days in Val d'Isere and Tignes. You'd have more accommodation options for a 7-night stay than a 3-night stay, you wouldn't have any time wasted with mid-week travel, the per-day cost of the lift ticket will be lower, and you'll still have an opportunity to hire a guide and do loads of off-piste skiing (the Espace Killy has loads of great off-piste terrain).
As for airport transfers, this site has a couple of pages of transfer companies http://www.valdinet.com/travel/transfers.html
. It takes about 3 or 3.5 hours to get to Val d'Isere from Geneva.