Wouldn't recommend Kew Gardens from Chertsey, it's a fairly long way with the traffic and not a good local parking situation.
If you want to visit the Royal Landscapes you can park near Blacknest Gate, walk round Virginia Water (about 4 miles), and visit the Valley Gardens, all for free (follow the marked 'Discovery Trail').
Savill Garden is in the same location, you have to pay to get in though. Personally I wouldn't go there, if you've not been to Wisley Gardens, which is much bigger and more interesting, for about the same entrance charge, and is easily accessible 1 junction along the M25.
If you like walking, there's Chobham Common, which is extensive lowland heath about 2 miles away. Weybridge is very close, a posh town where you could do a bit of window shopping, and walk to the Thames. There's also an out-of-town giant Tesco/M&S/Mothercare/Argos at Brooklands.
Staines is your best shopping choice, as it's only 2 miles away, and despite the Ali G associations, it's set alongside the banks of the Thames, and there's all the shopping you could wish for - the Elmsleigh and Two Rivers shopping centres with Debenhams, M&S, GAP, Mamas & Papas, Mothercare, New Look, Next, TK Maxx, etc. etc.