How would swimming in low gravity work? Sorry I know very little of actual physics, I'm not looking for hard and fast numbers just a general idea of how the mechanics would work.
For the sake of argument, say you were on a planet or moon that was mostly covered in ocean and had roughly the same gravity as say Mars, 38%. If you jumped into the ocean: 


*

*You'd be able to jump pretty high correct? Two or three times whatever you could on earth?

*How would the water react? I am assuming it would shoot up like a normal splash but much much higher then proceed to fall down very slowly?

*Would you be able to swim in the water? Similar to jumping, would the swim strokes take you farther than normal?

*If it were to start to rain on a planet with low gravity would the water droplets have to be much larger in order to fall?
 A: Interesting question. 
1 and 2 I agree with you.
3 - I think swimming would be similar to on earth - from the point of view of floating on the surface what counts is the density and our bodies and water have similar density - we are a bit less dense and float - swimming we force our bodies to go through water against the resistance of the water, which would be largely the same. - the only difference might be that the waves and splashes might be higher than we would expect for similar situations on earth, which could make it more tricky to swim on the sea.
4 - the size of raindrops depends on a number of factors - for large drops falling through air they tend to fall apart if more than 4mm in diameter on earth. In the lower gravity situation the 'terminal' velocity of raindrops could be lower and this might mean that drops could be slightly larger before they split up, but I do not think it would be a very big effect.  
A: 
1) You'd be able to jump pretty high correct? Two or three times whatever you could on earth?

Since you would have the same strength as on earth, the initial kinetic energy of your jump would be the same as on earth. Since your mass is the same as on earth, your initial jump velocity is the same as on earth. Thus the height to which you jump would be higher, since the height is given by
$$
h=\frac{v_0^2}{2g_{\tt mars}}
$$
and, per your specification, $g_{\tt mars}<g_{\tt earth}$.

2) How would the water react? I am assuming it would shoot up like a normal splash but much much higher then proceed to fall down very slowly?

You would shoot up much higher, and so would the water, for the same reason. The total time of both your jump in the air and the waters "jump" in the air would be greater than on earth. This is because the total time in the air is given by
$$
t=\frac{2v_0}{g_{\tt mars}}\;,
$$
and, again, since "g" is in the denominator, a smaller g leader to a larger total flight time.
You should be able to figure out the other two questions yourself based on the examples I have given. How does buoyant force depend on "g"? Does your density or volume change as you move from earth to mars? Et cetera.
A: Swimming would be nearly identical to a 1g planet, other than the splash being bigger. The forces involved in swimming are largely horizontal, so as long as there is some gravity to keep the water where it belongs you are acting against the viscosity of the water rather than the weight of the water. Might be a problem at very low g as you would splash away most of what you needed, but a planet with 0.2g would not have much of an atmosphere so the issue is moot - you would be skating rather than swimming.
If you go scuba diving you likely won't notice, other than ascending/descending quickly would require effort rather than simply adjusting your buoyancy.
A: In lower gravity, you could expect to swim faster
I am not answering the other questions as I do not have much more to
say which is not already said in other answers. But I do disagree with their conclusion that swimming would be
the same.
Regarding swimming, one would need a better understanding of swimming
motion to decide how much effect can be expected from low gravity. But
there is a point at which faster swimming should be possible. Here is
the analysis.
It is quite clear that static buoyancy does not change since it is
dependent on density ratio, not on the intensity of gravitational
attraction.
Drag, resistance of water to motion is also the same, so that it does not
change when buoyancy is unchanged, and the body is immerged to the same
level. Drag is the major factor that slows the swimmer, and it depends
heavily on how immerged he is.
This is so true that, for a while, swimming champions used the more
buoyant polyurethane swimsuits until they were banned in 2009.
However, there is more to immersion than static application of the
Archimedes' principle to control buoyancy. There are also dynamic
forces due to the motion of the body in the fluid that can induce
vertical forces in addition to drag, in the same way that an airplane
is submitted to a vertical uplift that maintains it in the air.
By controling the position of his body of of his hands, the swimmer can
create a vertical lift that can push her above water, as is done by
hydrofoils. This additional lift can be seen as extra buoyancy,
lifting a bit the swimmer outside the water, thus reducing the drag
of the liquid allowing for greater speed and/or less energy
expenditure.
This possibility is apparently not analyzed in physical studies of
swimming efficiency, but probably because lifting the weight of the
swimmer in Earth gravity is too demanding to be effective.
However, this need not be so in a reduced gravity. If we go to the
extreme, lifting the body is very easy in near zero gravity
(though you do want some gravity to keep air and water separate).
Without going to that extreme, there must be a gravity level at which
body and hand attitude can create some lift pushing the body outside
the water to reduce the drag of swimming in the water.  But it is
difficult to be more precise before we have an olympic swimming pool
on the moon to mesure the possibilities in field experiments.
