Why does our solar system revolve around the sun on pretty much a flat plane? Why do the stars revolve around the center of the galaxy on pretty much a flat plane?
Shouldn’t stuff revolve around these massive objects in a less tidy sort of way?
Well, because it revolve. Ever seen making of pizza base? If you take loose cloud of matter and spin it around, it will spread into a nice disk. Inertia and gravity. More interesting is question why it all revolve.
Elaborating, all these things aren’t revolving independently - they all interact gravitationally with each other. Imagine the Solar System consisted of the Sun, Jupiter and Saturn, and start the two planets out in orbits sharply inclined to each other. Whenever Jupiter and Saturn are near each other, Jupiter is pulling Saturn closer to its own orbital plane - if Saturn is “above” Jupiter, Jupiter is pulling Saturn “down”. Saturn, of course, is influencing Jupiter similarly. Eventually their orbits will more or less align.
Now generalise for n massive objects, and you’ve accounted for both the Solar System and galaxies.
If they started off in orthogonal orbits, wouldn’t they stay that way?
No, the gravitational attraction between two objects that are not in the same orbital plane will pull the objects out of their orbital planes. Objects in the same orbital plane don’t perturb the plane of each other’s orbit.
There’s also a collisional component. Imagine a disc of rotating rocky bits, and imagine a rogue orthogonally rotating asteroid. Every time that asteroid crosses the primary plane of rotation, it risks getting whacked into a more planar path. Over time, this helps keep things all rotating together.
Criminy, didn’t Laplace work all of this out, like, a quarter of a millenium ago?
Nobody reads classic physics texts anymore. 
Is the universal plane the same as the galactic plane? If so would the same reasoning apply?
Because planetary systems form out of disks of gas and dust orbiting stars. The cloud that the star forms out of collapses into a disk before the planets form. If you form planets out of a disk of material, they’re all going to be orbiting in pretty much the same plane unless something happens to change that. And it’s not easy to change the orbit of something large like a planet.
There is no universal plane. On the largest scales, matter seems to be distributed through the Universe in a pattern reminiscent of soap suds, and you don’t really get disk-like shapes for any structures larger than individual galaxies.