The Gilis are generally drier and hotter than Lombok or Bali. The dry season is from April to October, the rainy season from November to March (but it rains less than Bali; we visited in February and had plenty of warm, dry weather). The peak tourist seasons are July-August and December-January.
The Gili Islands are situated between Bali and Lombok. Although Lombok International Airport is closer, Bali's Denpasar Airport is better served by international flights. From either of these, there are fast boats to the Gili Islands. To Gili Air, it is approximately 1 hour 30 minutes from Bali's Padang Bai port, or 30 minutes from Lombok.
You can also fly to Jakarta's Soekarno-Hatta International Airport, then travel overland or take a connecting flight to Denpasar or Lombok.