Would a more significant curvature of spacetime relative to earth (i.e. time on jupiter) effect the aging process? Basically would you age at a slower rate? Or age faster on the moon?
Einstein's general theory of relativity explains time dilation. Wikipedia provides a concise summary:
time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other, or by being differently situated relative to a gravitational field. As a result of the nature of spacetime, a clock that is moving relative to an observer will be measured to tick slower than a clock that is at rest in the observer's own frame of reference. A clock that is under the influence of a stronger gravitational field than an observer's will also be measured to tick slower than the observer's own clock.
As one experiences more gravity, time seems to flow slower compared to (say) Earth. That means that "standing" on Jupiter, which isn't possible due to the gaseous surface, will cause you to move through time faster; but you do not age slower. Suppose your lifespan is 80 years. On Jupiter you still would live 80 years, but the time that has passed on Earth during your 80 years would be slightly more than that.
The gravitational time dilation on the Moon is the opposite: the weaker gravity means that an Earth clock ticks very slightly slower than a Moon clock.
The faster one travels, the slower you experience time. For example, if a person were traveling at $0.99c$, and assuming that a human lives for exactly 80 years, the person would still live 80 years, and life would be normal from their perspective.
However, an outside observer would see that the fast human would live for thousands of years, as the observers are not experiencing time dilation. Let me provide another example.
In the movie Interstellar, astronauts orbit a supermassive black hole. When they visit a planet in the vicinity of the black hole, though they think they only have stayed there for a few hours, over 20 years have passed. This is because they are under a massive spacetime curve caused by the pull of the black hole.
On the other hand, if a person was on a low-gravity body, like the Moon or Mars, time would be slightly faster than if they stayed on Earth. The 80 years that they would live would be normal to them, but to someone watching on Earth, their life would be slightly shorter than 80 years.