It sounds like you're going to make a really small trailer. Can you cut both sides from one 4x8 sheet?
I believe Doug used a product called MDO (medium density overlay) for the sides of his latest trailer. You might look into this product as it appears to lay fairly flat compared to most exterior grade plywoods and it has a phenolic finish on the surface to begin with. This product can also be purchased pre-primed or pre-painted for a few bucks more, much of the prep work has already been done. Here's a link to my local supplier so that you might investigate these materials:
http://www.alumapanel.com/products.cfm?step=1&lines_ID=310
If you don't mind spending some money, you can get 4x8 sheets of exterior grade ply with painted aluminum factory bonded to the surface. Might be a nice way to go. You could consider just using painted .040 aluminum sheet for your roof and have an all metal exterior, pretty much ready for your color of choice...without a lot of surface prep that you would have with wood. Costs a bit more up front, and I have no idea what your budget is...but this is a small trailer so the raw materials might not kill the deal.
http://www.alumapanel.com/products.cfm?step=1&lines_ID=290
These products are available from sign supply distributors who supply the local sign makers with something to work on.
As far as the "tileboard" that HD has...its not much. It might be able to be used for an inexpensive trailer but the stuff is pretty marginal and doesn't have much resistance to water, even though it sounds like it would, "tileboard" and all that. BTW, the low cost white HD tileboard usually has a textured backside, I'd rate it one step above cardboard. Lowes has a similar, low cost white tileboard that is much more like a tempered hardboard, with a dark color hard back side. Look at the product at each store and see the difference. I buy maybe a dozen sheets of this stuff a year to make stencils for painting, and I'd drive forty extra miles to by the Lowes over the HD. I think you'd be much better off with plywood. 1/8" birch might get the job done for a small trailer like yours, unless someone sat on it.
