First of all congrats on the the DSLR purchase! (I remember the excitement of my first SLR) If you haven't already, you will eventually get the "lens" bug and want to add all kinds of glass.
My number one advice is to not buy too much too soon! Take your time, learn what you have and figure out what your needs are/become based on time and experience.
Here are some options I would seriously consider:
(I didn't do the math but this should be close to your budget)
Rent (for your Malaysia trip)
70-300 or 70-200
The 10-22 plus the 24-105 gives you the perfect walk-around range plus some telephoto. This covers landscapes to moderate telephoto on a crop body. Don't under under estimate the need for a wide lens on a crop body such as the 60D. The only down fall to this combination is swapping lens--but isn't that one one of the benefits of an SLR?
Haven't been to Malaysia or shooting in a rain forest but I would expect the RF is heavily compressed vegetation. I'm betting the 24-105 (38-168mm FF) is more that sufficient reach for a crop body in the RF. Plus the 24-105 is weather sealed. The 10-22 is not sealed so be carefull.
Now for a telephoto zoom, I would rent and not buy one at this time. I'm assuming this will be used outside the rain forests in more open areas (otherwise I'm betting the 24-105 will do the job). I'm a huge fan of the 70-200 IS lens like many others. For this trip you might want something longer than 200 but when you get back home you might find that a lighter faster 70-200 is the best choice. So rent what you need for your trip and based on your experiences, you can buy a telephoto zoom later.
If you are set on buying a tele zoom now, go for a 70-200L IS--absolutely wonderful lens, IQ, build, etc. I have the F4 version. Between the 2.8 and F4 versions, it comes down to speed or weight, and of course cost.
And if you are set on the 15-85 or 17-55, those are both great lens. Neither are sealed and expect excessive distortion at 15 on the 15-85 and some zoom creep--I use to own the 15-85 but not the 17-55. Personally I'd get the 10-22 and 24-105 and rent a tele zoom
Don't forget HQ filters on your lens for moisture and physical protection. I use B&W UV filters