Chains and casette are rusty I used to live in Australia where rain is scarce. I've recently moved to SEA where it rains almost everyday in the past month. Hence, the road conditions are less favorable as they are always damp regardless the time of the day. 

Recently, I've noticed rust on my chain and the surface of my cassette. This did not happen in Australia as I clean my bike after every ride, and a thorough wash every week. Hence, I'm not sure whether the weather here is the pain in the ass or there's just something wrong. I've always been using dry lube because I've bought bunch of them in Australia. Will that be a significant cause?

If so, what should I do with the rust? They could be scrubbed off fairly easily but there are still some left between the ridges. Should I bring it in to the LBS or just shrug it off and try to get some wet lubes and observe for some time?
A great rule of thumb is "Dry lube for dry conditions, wet lube for wet conditions". 

A bit of surface rust on an otherwise well maintained cassette and chain is only a cosmetic problem. If this is a concern, clean the chain, lube with a wet lube, wipe dry. The trick is to 'over lube' the chain and then wipe off the excess. I might be tempted to leave wiping the chain for a day or two to let the lube migrate around the links. 

As far as the cassette, give it a light spray of WD40 after cleaning. Don't get too much on the chain as its not a lube and you don't want it displacing the lube you just put on.