There are all sorts of things you can do to get more manageable video sizes. The first is to shoot in a size that's no larger than you need. You're probably at 1920x1080, which is huge. Unfortunately, you don't have any options with the built-in camera. But there are a number of great, inexpensive camera apps that allow you to shoot at lower resolution. I use ProCamera 7, which has all sorts of great advantages in addition to lower resolution video (not affiliated, just a very satisfied customer). I rarely shoot in larger than 720p.
Otherwise, you can accomplish your goal in Handbrake, which I use, too. You should see a number of presets on the right that should get you into the ballpark. Then, you can adjust video quality (the "Video" tab halfway down on the main screen. There's a slider for "Quality". Try adjusting this to larger numbers for reduced quality and smaller file size. You might find a quality value of 22 is good enough. 26 is pretty marginal, but I'll use that to get a TiVo'd TV show onto my iPad for a one-time viewing. When I rip a DVD, 20 is plenty good enough quality.
On the "Picture" tab, you can adjust frame size down if you'd like. Set "Anamorphic" to None, and select Keep Aspect Ratio. That will allow you to decrease Height from say 1080 to 720, and the Width will adjust accordingly.
Quality and Frame Size are all you need to mess with to get a much smaller file size.
