What is the difference in pressure of an object floating VS being pushed down in water VS being held by a string in water?
I assume that you are talking about the pressure of the water on the outside of the object.
That is only a function of depth below the surface.
At the surface, pressure is the pressure of the air, one atmosphere.
As you submerge, the pressure goes up linearly with depth (assuming constant water density which is a reasonable assumption but watch out salt water is denser than fresh)
so p = Patm + rho g h
where rho is the water density and g is acceleration of gravity and h is distance below the surface.
So for example if floating, the presssure at a point on the bottom of your ship would be Patm + rho g d
where d is the draft (depth of bottom from waterline) of the ship
As you push down, d becomes bigger and so the pressure at the bottom of your ship goes up.
As you totally submerge, d gets bigger and bigger and therefore the pressure keeps going up.