DIAGNOSIS Some surface rust on the flavorizer bars is normal. Rust on the cook box interior bottom is also common from moisture and grease acids. It's not a structural issue yet.
FIX For the flavorizer bars, scrub them with a grill brush while warm. For heavier rust, use a wet pumice stone. For the cook box floor, scrape out loose debris and rust with a putty knife. You do not need to get it to bare metal. Just remove loose material.
IF THAT DIDN'T WORK If rust is perforating the metal (actual holes), the part is failing. Order a replacement cook box or flavorizer bars from Weber. They are designed to be replaceable.
HEADS UP The best prevention is to run the grill on high for 15 minutes after cooking to burn off moisture and grease. Use a fitted vinyl cover.