It would be great if we would send one or two such probes out per year, in different directions. We could find so many interesting things. There are currently only two.
The Voyager probes where only able to escape the solar system because Jupiter, Saturn, Uranus, and Neptune where aligned, and they could use the slingshot effect to accelerate when passing by them. These planets will not be aligned in a similar way until the 22nd century [1][2].
The "slingshot effect" or gravity assist involves a vessel robbing angular momentum from a large body.
If you launch a vessel towards a planet (so that you come near but don't collide with it), then in the reference frame of the planet your vessel will leave with the same velocity going away from the planet as the velocity of your approach. However, since the planet was moving with respect to the Sun, you will have gained or lost velocity with respect to the Sun, depending on whether your space craft approached from behind or in front of the moving planet. Typically in these assist maneuvers, you wait for a large planet to be a bit ahead of the Earth's orbit. You launch towards that planet, and the gravity assist accelerates the space probe. Some of that acceleration will be tangential (mostly useless to you) but some will be radial, increasing your velocity from the Sun.
The Voyager probes took advantage of a once-in-every 180 year planetary alignment, with all of the outer planets mostly aligned. See http://en.wikipedia.org/wiki/File:Voyager_2_path.png for a good illustration. All the outer planets are traveling in a counter clockwise direction, and you can clearly see the outer planets accelerating Voyager 2 tangentially. It should be clear that there will be a radial component to that acceleration as well.
I don't think you can use the Sun in these circumstances; it doesn't have any angular momentum for you to steal.
Incidentally, the discovery of this method is sheer brilliance; without using gravity assists it would have taken prohibitively long and required a prohibitive amount of propellant to reach the outer planets.
A theoretical and probably ridiculous question here, but... does not the probe thus reduce the planetary orbital speed by the same amount (relative to mass, I guess), that it gains?
Momentum is conserved in the system (total momentum of probe and planet is always the same). Momentum = mass x velocity, so yes, the planet will slow down, but it's a ridiculously small amount. Think of shooting airgun pellets at a tank barelling down on you to slow it down, but even less effective.
The slingshot effect lets you add or subtract a fraction of the orbital motion of a planet you closely fly by. The key is that the speed relative to the planet will be the same going in as coming out, however the angle of the trajectory of the spacecraft relative to the planet's orbit can be different. If you want to change the plane of an orbit the slingshot effect can still be worthwhile, since it can help kill some of the speed in the ecliptic plane, but it won't help adding to the speed perpendicular to the ecliptic.
Could you not build up a lot of speed in the ecliptic plane with the slingshot effect, and then use the gravity of a planet or the sun to deflect the orbit perpendicular to the ecliptic? (I mean in practice this maybe be unlikely to be feasible, but in theory it is with the right planet arrangement)
Yup, you can do that, but the faster you go the less you'll be deflected, and you won't get a speed increase as you would with a normal planetary assist.
That's my point exactly. Don't wait. Send probes out every year. Because it takes so long for them to get to these places. A few may get lost on the way. Some instruments may malfunction. So better send out more. What would that cost? Half a billion dollars a year? Probably much less, because construction would get more efficient.
