I've put the mods in the /mod directory, and i'm getting the following crash report.
I've tried two other versions of Project Red, but for some reason it always crashes.

The following is the crash report for the latest version of Project Red:

http://pastebin.com/8AYsfxsF

You've somehow corrupted something. Delete all the PR jars and redownload.

The problem is you didn't bother to read the instructions on the downloads page. Mechanical won't work without Integration.

MrTJP, i don't know what you mean:

http://i.imgur.com/5cCPiWv.png

I have both mechanical and Integration?

kenzierocks, i'll try redownloading

Well, thats very strange. It doesn't show up in the logs at all. I don't think its being loaded.. (In which case, all I can suggest is redownloading it)

Don't know why, but I redownloaded and was able to get it to boot up.
Thanks kenzierocks and MrTJP for your time!!

Definitely a failed download. ProjectRed-1.7.10-4.5.10.61-Integration.jar is 3.33MB, not 20KB as listed in that screenshot.
In future, check the FML log too, it will say something like "Cannot read from jar file"