I'll start off with the best place to eat- I got this recommendation from here and ate there 2 out of the 3 nights we had there. It has to be Jalan Alor. Its a street of full of restaurants and stalls which spill out on to the road. Look for a busy place and choose it. The food we had was fabulous. Great people watching place as well

Shopping
Sungai Wang - sells everything with prices cheaper than chinatown (an experience not to be missed),
LowYat - just behind Sungai Wang sells everything computer based.
KLCC & Pavillion - upmarket brand name shops (fixed price only no bargaining) but good choice of shops.

Best times to shop are weekdays when you almost have the shopping centres to yourselves and get good service, I would recommend avoiding weekend shopping if possible (especially Sungai Wang and LowYat as it seems like everybody in KL is shopping at the same time).