Part of your question is "How fast does gravity travel?" This is a matter of debate, particularly because we haven't observed a rapid change in mass like the scenario you described.

Classically, gravity can be thought of as a distortion in space and time. Think of a bowling ball on a mattress - a marble placed nearby will roll toward the bowling ball because of the surface of the mattress curves. From that perspective, the gravitational effects of removing the Sun would strongly depend on how that distortion was removed. (If the distortion wasn't removed, for example the sun were replaced with an equally massive black hole, there wouldn't be an effects on the planets' orbits.) I think that moving tangentially away along straight paths is the most likely scenario. If the Sun's mass were slowly reduced, the planets might spiral outward as their orbital radius increased with time.

Settling around Jupiter seems less likely - while massive for a planet, Jupiter has only 1/1047 (0.0955%) the mass of our Sun and is only 317 times the mass of Earth, so the mass ratio between Jupiter and Earth is smaller than the Sun and any of its planets, which leads to weaker forces. Plus, if the other planets move tangentially away from their current orbits, it is most likely they would move away from Jupiter too fast to establish stable orbits. This is an interesting question and I think you could explore this with some calculations on scratch paper using the universal law of gravitation:

F = G*m₁*m₂/d²

How does the force compare if m₂ is Jupiter instead of the sun? At what distance would an Earth rotating around Jupiter have the same force as one rotating around the Sun (at our current distance)?

Jupiter's mass is not enough to hold the solar system together. If it were, then all of the other planets would be moons of Jupiter, which would in turn be orbiting the sun. Yes, removing the sun would cause the planets to zoom off at tangential "orbits".