You can do most of that within Vegas easily. The sequence was likely made in After Effects, however, judging by the look of it.

Use track motion. There are some scripts for Vegas that simulate camera shake, I think Vegassaur has some you can use. that will give you much of the camera motion effects you are after. Then you can experiment with various transititions. and even making some custom transitions. The video tutorial below will point you in the right direction. From there it's just a matter of making some shapes that you can animate in and out of the scene using things like track motion combined with transitions (try wipe transitions on solid color media, then play with changing the size and shape of the media using the track motion GUI to create lines that animate in and out of the scene). Once you have a line that animates in and out, hit it with a mirror effect or two to get some nice geometric patterns that animate in and out of the scene.

Vegas also has a bunch of other media generators you might find helpful for creating backgrounds and such, and there's always the buying pre-made backgrounds route.

Then overlay some text and use various transitions or effects to animate it in and out, apply the same shake effects to the text, and you should be getting pretty close to what you see there.