I understand the basics, protons have a positive charge, neutrons have no charge, and the electron has a negative charge. But looking at the lines of force from a proton, they flow outward and push each other away. But, the electrons flow inward or towards themselves. How does that make them repel? Wouldn't they be more neutral towards themselves? I understand that the lines of force cannot cross as well. I'm sure this is an easy answer for someone.
The lines of force represent the direction a free positive charge would move if one was present. The reason the lines of force are in the outward direction from a proton is because a proton will repel a proton, and thus move outwards. The reason the lines of force are inwards for an electron, is because an electron would attract a proton, thus the proton would move inwards.
So to summarize, force lines are defined to be in the direction a POSITIVE charge would move, if one were present. This is just a human convention/definition.
An electron has the opposite charge properties to a positive charge, and a free electron will move in the direction that opposes the force lines. So an electron will move away from another negative charge, and towards a positive charge.
The force experienced by a charge is $F=E\cdot q$. Let electron number 1 be in some point in space. Its field lines are directed towards it. Now put another electron near it. Since the second electron's charge is negative, the above product implies that the force the electron feels is in the opposite direction than that of the field line - namely, away from the first electron. So the electrons do repel each other.
Field lines do not 'cross' each other because if you have two fields generated by two charges, you can just add the fields together to get the total electric field in space. Field lines can, on the other hand, converge/diverge - and any point where they do so is a point where an electric charge exists. Just like you said - they diverge away from a proton, which is a positive charge, and converge into an electron, which is a negative charge.