You're going to need Perian to work with subtitle files in Quicktime. If you don't have it, head over to Perian.org to download and install it.
We need to make sure Perian is loading subtitles, so open up preferences and click Perian. The "load external subtitles" item at the bottom should be checked.
Go to the folder your movie and subtitle file is in. Before you open the video in Quicktime, make sure both files are in the same folder and have the same filename. If you have .sub subtitles instead of .srt subtitles, you're going to need to convert them. You can use Sub2Srt found at the link in the Resources section below. If you can't get that to work, do a Google search for subtitles and find a .srt version of the subtitle file you need. It's not as complicated as it sounds, just give it a try.
Open up the video file. Play a section of it and make sure the subtitles are showing up. If everything's running correctly, go to File in the menu bar and hit "Export..."
The "Save exported file as..." dialogue should pop up. Name the file and select your Export type. If you're unsure what type to use, I recommend MPEG-4. Click the Options button.
Change your desired export settings. If you're unsure what to enter, just put in the settings pictured here.
Click OK and hit Save. The exporting progress bar should pop up. It will take some time to finish, so be patient! When it's done you can delete the subs and other video file. Your main mp4 will include both the video at the subs.