Hi Nolanp2. I don't think visualising tensors helps to understand them.
Thinking about geometry might help. You mention force fields, so I'll give you an example of how and why tensors are used. Consider the equation of motion of a charge in an electric field,
m.a = e(E + v x B ), i.e. mass times acceleration is electric charge times E + v x B, where v is the velocity vector and E and B are electric and magnetic field vectors. x is the vector cross-product.Note that this equation is really 3 equations, one each for x, y and z.
This formula is fine but it is not relativistic. It won't work if you put it in a moving frame of reference.
To make the equation obey special relativity, we go to four dimensions,
t,x,y,z, with stipulation that the time dimension has a negative sign, and
that we multiply all times by c=velocity of light to make them distances.
We can now rewrite the equation in tensor notation,
ma^{\mu} = e F^{\mu\nu}v_{\nu}
which is four equations since \mu can be t,x,y,or z. F is the EM field tensor ( see Wiki for instance).
But the point is that this equation is relativistically covariant, i.e it transforms properly between inertial frames. It's also very elegant.
To do a calculation from this formula, say for x ( mu=1), we expand the tensor to give,
ma^{1} = e F^{10}v_{0} + e F^{11}v_{1} + e F^{12}v_{2} + e F^{13}v_{3}
Note that all the indexes are numbers, and we can plug in the values of the components to get an algebraic differential equation. The tensors are gone.