When you open HandBreak it should pop up a window asking for the source (i.e. click on the DVD-disk). It scans the disk and it then shows you in the Title drop-down list which episodes/sections it found.
There should be one for each episode, but there might also be some that are like 10s long (like an intro/background etc.). Just select the first episode you want to rip, use the Presets (if they are not visible, click the Toggle Presets in the toolbar) like if you're going to play it on an iPhone, select the iPhone & iPod Touch preset. Then just name the episode and press "Start"
If you want to rip the entire DVD in a row, just use the Add to Queue feature to build up a queue before you hit Start.
That's basically it. You can go more into depth with picture quality, subtitles etc, but the presets works really well. Oh and if you're unsure if the episode (from the Title drop-down list) is the one you want, you can click on the Picture Settings... button on the lower right, this will show you 10 screen shots from that episode, so you can easily check that you've got the right one. (if you mess with any of the settings in that panel, just hit the Preset of choice again.)