The bulk of the radiation we have to deal with around Jupiter would be held within its Van Allen belts. If Earth settled into an area between (or beyond) the radiation belts, it would be fairly safe from those radioactive waves. Also, we have our own magnetic field that would provide us with shielding from Jupiter's radiation. With the amount of luck you are calling for here, landing in the right spot wouldn't be too hard. We are currently working on maps of Jupiter's radiation belts in an effort to figure out where to send probes...but as we know from Earth's Van Allen belts, there are gaps between them.
Space is big™, there is Plenty of space around Jupiter for Earth and Mars to find an orbital path where they won't actually impact one of Jupiter's moons. However, getting both planets to orbit Jupiter in a stable fashion would be harder. Simply because Earth and Mars would be gravitationally pulling at each other on a regular basis... So Earth and Mars would have to orbit at very different distances from Jupiter, or they would destabilize each other's orbits. See this other question for math and numbers on how to more specifically position a 'habitable satellite' around a gas giant. But, all said, we're going to need to be a pretty good distance away from Jupiter in order to have a safe, stable orbit.
Getting out of Earth's gravity well wouldn't change much as far as clearing the atmosphere goes. The biggest difference would be the location of the Earth-Jupiter Lagrange points, particularly the L1 point (the point at which, if you stopped, you'd fall towards Jupiter instead of Earth). Once you hit that point, you could use orbital dynamics to throw yourself at Mars by slingshotting off of Jupiter (make sure you have good radiation shielding, since you'll probably hit a Van Allen belt. Jupiter's can kill an unshielded human in minutes). Since the Earth/Jupiter L1 point would be closer to Earth than to Jupiter, it would require less energy to reach than it would require to otherwise travel to Mars. Somewhat. Since force of gravity is related to the distance between two objects squared...most of your energy is expended at close range to whatever gravity well you are trying to leave, rather than once you are further away.
Also, travel would be much easier one way, and harder the other. Simply because Earth and Mars would have to orbit at different distances from Jupiter in order to be stable. So, to get to the one closer to Jupiter, you'd simply have to 'fall' towards Jupiter, and then stop at the right place. To get to the outer planet, you'd need to burn against Jupiter's gravity well as well. Again, this wouldn't be a huge difference, because of the distances involved...but still noteworthy.
Some theorize that Jupiter is one of the reasons that our inner solar system is habitable. It acts like a big gravitational linebacker and intercepts long period comets, often flinging them clear out of the solar system rather than letting them come into the inner system where Impact is a risk. With us sitting in orbit around our gravitational linebacker, we are likely to see more things hitting our atmosphere as Jupiter accretes space junk. Not to mention the fun to be had with any asteroids it picked up on its way through the asteroid belt.
Tides on Earth are also going to be interesting. We have another major player in the tidal-force game added in, in addition to our moon and the sun. Additionally, our moon (if we get to keep it) would pick up a more eccentric orbit around us as it is tugged on by Jupiter, and would result in varying strengths of tides as the moon drifted closer and further to us in our orbit. And depending on our distance from Jupiter, and the tidal forces exerted on us, we may also see a rise in volcanic activity.
Weather would also get interesting, as (with Jupiter) our orbit around the sun would become more eccentric, leading to more extreme seasons. I'm not sure how Earth's energy budget would be impacted, with the addition of the energy and forces produced by Jupiter (some of its moons are internally heated to above 0C simply from Tidal Forces) but it would likely change.
It would take a really long time for this to happen, but eventually, days would get weird. The question I linked above includes equations to determine Tidal Locking...which would happen eventually. We would be tidally locked to Jupiter (so on side of the planet would always face it) and so our day would end up the same length as our orbital period around Jupiter. This would take much less time than is projected for Earth to become tidally locked to the sun.