The solar system is indeed pretty much a flat sheet, with the major planets all orbiting in a very thin plane surrounding the Sun. Part of the reason we don’t tend to send spacecraft in the 'up' direction, out of this thin plane, is simply that there’s not very much...
The solar system is indeed pretty much a flat sheet, with the major planets all orbiting in a very thin plane surrounding the Sun. Part of the reason we don’t tend to send spacecraft in the 'up' direction, out of this thin plane, is simply that there’s not very much there! Now, that’s not to say that there isn’t anything out that direction, but you have to travel for a while before you reach it.
Closest to the solar system, but at its outermost fringes, the orbits of the objects in the Kuiper Belt deviate from this extremely flat plane, but still tend to orbit mostly in a disk surrounding the Sun. Instead of an extremely flat plane, you have something more like an inner tube - inflated, with some vertical height to it, but still mostly lining up along the plane of the rest of the major planets. This area is where Pluto falls - its orbit is tilted out of the plane of the major planets by 17 degrees, but it’s not so far tilted out of the plane of the rest of the planets that it’s really traveling overhead the other planets.
What you do get overhead the planets is a much more distant object, the Oort Cloud. This is a reservoir of comets, incredibly distant from the Sun, which are arranged in a roughly spherical distribution around the Sun. The objects out here are small, dimly lit chunks of ice and rock, and so far from the Sun that they are extremely difficult to observe, even with high end telescopes.
If we travel further away, and look for even more distant objects, then suddenly we run into a proliferation of stars within our own galaxy which are 'up' above the plane of our solar system. Part of this is that the galaxy is much thicker than the solar system, and so even if the plane of the galaxy and the plane of the solar system were perfectly aligned, we would see stellar neighbors of our Sun, both above and below our solar system. However, our solar system isn’t perfectly well aligned with the Milky Way galaxy- those two are off from each other by 63 degrees. What this means is that we see far more stars 'up' or 'down' out of our solar system, as we look through part of the densely populated disk of the galaxy, than we would if we were looking directly 'up' out of the plane of the galaxy.
The constellations are a good example of things that exist 'above' our solar system. Our planet spins on an axis that’s tilted by 23 degrees relative to the plane of the solar system, so looking at the stars at the exact North Pole isn’t quite pointing us in the right direction, but it’s pretty close! If you find the North Star, Polaris, and then wander about twenty degrees (about two fists, held at arm’s length at the sky) away from the path the Moon travels, now you’re pointing 'up' out of the solar system. The constellations in this region of the sky are plentiful. This is roughly where the cup of the Little Dipper is, though exactly 'up' is in the much fainter constellation Draco (illustrated at the top of the article). The stars that hang 'above' our planet remain roughly stationary in our night skies as our planet rotates beneath them - if you’re far enough North these stars will never set.
Since the direction of 'up' away from the solar system doesn’t also point you directly out of the Galaxy, if you want to face a direction that aims you at the center of the galaxy or 'up' out of the Galaxy, you need to point yourself in a different direction. The center of the galaxy is in the direction of the Sagittarius constellation, which looks a bit like a very pointy teapot in the sky. To point your face out of the galaxy, you must aim yourself at the lesser-known constellation Coma Berenices, which is surrounded by other constellations you’ve probably heard of - Virgo and Leo border it, as does Ursa Major (which contains the Big Dipper). If you take the last three stars in the curve of the handle of the Big Dipper, and imagine them creating a long, pointy pizza wedge, you’ve gotten pretty close to Coma Berenices. What's in that direction? Not much that's visible to the naked eye, but if you have a well-equipped telescope, you run into the Coma Cluster, a very dense collection of galaxies - an environment very unlike the galaxy our own Milky Way finds itself within.