My mov files are too huge, even if they're only 10s long

Start here: User Guide

Check out the Export from After Effects course about halfway down the page.

If you need an alpha channel and you do not have an expert's understanding of video formats, standards, and compression just use the Lossless with Alpha preset in the Output Module. You are going to be stuck with large file sizes because the formats that support alpha channels have at a total of 32 bits of information, every frame is unique and the color sampling is higher than it is with delivery formats. 

If you want the best quality for the least file size and the greatest compatibility, use the GoPro Cineform codec and learn how to set it up properly.