So after 8 hours of research, trial and error, I found a successful workflow for .mov footage from my Canon EOS Rebel T3i.
The first part is that you can't import .mov "Apple / H264" files with Lightworks without purchasing the "Pro" version. The pro version costs $60 per year (which is actually a good price for what you get). So I bought that and was able to import my .mov files.
The next problem I had was that playing the imported footage was horribly slow and choppy. I wanted to "mark and park" to get clips added to my editor but when I would play the video, it would only play a few frames and then get horribly choppy (audio would still play). By the way, I have a laptop with an Intel Core i5 (2nd gen) and 4GB of RAM.
After some research, I found the solution to this was to import the video while transcoding it to AVI format with "DVCPRO100". This made the video playable within Lightworks for editing.
Next I had some real problems exporting my edits into a codec/format that would actually work. Some formats would just error out (post an error and exit). Others would succeed, but the final video would not have audio (I think this happened with MP4). Finally I found that exporting as "Quicktime Movie" (not to be confused with the "MOV" option!) worked. It worked with the following options:
H.264, Quality = High, Key framerate = 24
Frame Recording = Yes, Encoding Mode = Multi-pass
Sound: Format = Uncompressed, 44.1 kHz, Channels = 2
For the other Lightworks export options, I chose:
HD 1080, 29.97 fps, and Progressive button checked.
I sure hope this helps other people save 8 hours of frustration! With this workflow, I can now be efficient and productive and make some great videos.