I'd like to put in a good word for Shotcut. (It's also good for making gameplay gifs, though you will need separate screen capture software)
I think that while WarpZone's advice mostly holds, that your trailer could be mostly gameplay footage, but probably in clips of no more than three seconds, showing off important gameplay features or particularly pretty scenes. Then intercut that with some text on a black background. You can get a lot of milage out of that.
I use OBS to capture gameplay, it's fiddly, but has the good features and plays nice with my hardware.
And if you want some free music for your trailer, CCMixter has a huge collection of it.