Use the Add Folder command in iTunes and it will copy it into iTunes under movies.
Right click the movie in iTunes and chose Information, that will tell you about the resolution, bit rate and so on.
Provided you used one of the Handbrake presets for iPods, and didn’t fiddle with it, it will play in itunes with no trouble and it will transfer to your iPod.
If you didn’t chose an iPod preset and let Handbrake use its default normal then you’ll have another step.
After you right click on the movie chose Advanced from the top of the iTunes screen and click on Convert iPod/iPhone to get the version for that.
That will leave you with 2 copies, one will look pretty good on a large screen and the other will look good, and run, on your iPod.
When you’re sure you’ve got what you want you can delete the movie from the folder you used when you converted it.
Time to read all the Handbrake documentation while you’re waiting for the conversion to complete.
edit : Remember, the movie is first put into iTunes before you can put it on your iPod.