Their orbits are planned out before the mission to begin with. If they weren't no mission would be possible.

The orbits of the planets in our system are well studied, so we know their shape around the sun and the speed the planets have along each part of their orbit, so modelling their orbits into the future is relatively simple. If you've ever wondered how we know before hand when conjuctions will happen or when a comet will pass by or how astrological predictions are made for future dates, that's how. (Astrology isn't a real science but it relies on our good understanding of our solar system).

We also know their mass and their gravitational force, so complex calculations are made that can chart a spacecraft's exact course through space which also determines how much fuel the spacecraft itself needs for maneuvers and how much of its course will rely on the gravitational pull of other celestial bodies. Space is mostly empty, and vast, so the chances of a random piece of debris or asteroid hitting the spacecraft are not zero but they're highly impropable.

Nowadays these calculations are carried out by complex computer modelling but it's not a simple calculation that one can simply write out here. But what you can do is get Universe Sandbox, which is a game that models our solar system fairly well and allows you to change variables or simulate missions which helps visualise and give a much more intuitive understanding of how these things work. In fact there's a whole host of such software but this one I think is a great one for people interested in learning about our solar system, space and how missions work