Planning missions in New Horizons will be somewhat simplified, but still pretty involved. Spacecraft speed with be measured in AU/yr, and distances is AU. For reference, the distance between the Sun and Earth is by definition 1 AU; the distance between the Sun and the Asteroid Belt/Ceres is 2.5 AU, and from the Sun to the Kuiper Belt/Pluto is about 40 AU.
To travel between two bodies, you must travel the distance of the larger orbit. For instance, if you're travelling from Jupiter (5.2 AU) to Saturn (9.5 AU), you must travel 9.5 AU. This is because it's the average of the distance between the two bodies and I don't want to track the positions of the planets. In the best case, the distance is 9.5 - 5.2 = 4.3 AU, and at worst, the distance is 9.5 + 5.2 = 14.7 AU. In addition, once you reach a body, you must lose all speed (via aerobraking, parachuting, or propulsion) to land, which makes soft landings on barren bodies very difficult. Even if you're not landing, you must lose most of your speed if you want to change direction and go to another planet.
That is, unless you use a gravitational slingshot (also known as a gravity assist). I'm going to randomly generate which years allow which gravitational slingshots, but a good slingshot around Venus can net you almost 7 AU/yr. Unfortunately, these only allow you to go to a single body, and if you want to keep your speed, you have to slingshot AGAIN around that planet. This is what Voyager 2 did: it flew by Jupiter, used its gravity to slingshot to Saturn, then slingshotted around Urstar fish, then finally flew by Neptune. Of course, this isn't necessary if you plan on skipping planets.
If you're having trouble imagining gravity assists, here's an anology (from NASA and Wikipedia): imagine a train running at 100 km/h. Now imagine you throw a tennis ball at the train with a speed of 50 km/h. When they collide, the tennis ball will bounce off the train, still with its 50 km/h velocity. However, now the train's velocity is added to the ball, so really the ball is going at 150km/h. (Technically, the ball could get up to 250km/h, but it's complicated).
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