If the asteroid belt is a disc, couldn't you go above or under it rather than through it? Or does it form a sphere around the terrestrial planets orbits? Edit: I know the chance of hitting an asteroid is extremely low, so just think of it theoretically
Yes, you could go "over" or "under" the asteroid belt. However, plane changes are expensive, and as pointed out in the comments, the asteroid belt is not very dense (average distance of 600,000 miles [1 million kilometers] between objects) so there isn't much to avoid.
Purely propulsive plane changes are expensive, but a much smaller adjustment followed by a swing-by maneuver over the pole of a planet a bit like Voyager 1 did at Saturn, (which in this case I guess would have to be Venus, Earth or Mars), might be much less expensive.
Or you could just leave Earth orbit in a direction that doesn't happen to be in the Ecliptic plane to begin with.
In either case if the mission were to visit a body in the ecliptic, you'd then have to make a large $\Delta v$ maneuver to get back to the ecliptic, and a second swing-by of perhaps Jupiter to stay there.
However, if the mission were to a body that wasn't in the ecliptic to begin with, then the rocky-planet swing-by's or initial out-of-plane trajectories would be all you'd need.
It would be a bit like sailing your ship from Spain around the cape of Africa rather than through the Mediterranean sea to get to Egypt just because you were worried you may hit off Malta or Sicily.
There would be significantly more to avoid within a significantly smaller space in low earth orbit than out in the asteroid belt which is extremely empty compared to the image movies and tv shows have presented us with.
The cost of going around or "over" it would mostly be measured in time and would be a completely worthless endeavor. You would also have to figure out how to then correct your course as space navigation is not the same as on earth where you apply force in the direction you want to go. 99.999' % of the force acting upon any object in space is gravity and propulsion systems are not used to cover distance so much as to adjust trajectories so that gravity and angular momentum propels the craft in the desired direction. As for where you are going that also has to be worked out ahead of time as everything is moving, you need to aim for where the thing will be at the time you get there rather than aiming at where it currently is. If you take the time to go around something then the thing you are aiming for will have moved. You cant travel to another planets location, you have to intercept it at a certain point along its path.
The asteroid belt ranges from 2.2 to 3.2 astronomical units (AU) from the Sun and is 1 AU thick, see. So it is a very long way to go over it and too much fuel would be necessary. All the very sucessful spacecrafts to the outer solar system would have been impossible if they had to use a trajectory over the belt.