How do particles get their charge? 
*

*How does an electron get its charge? 

*And how can it maintain that charge for very long (infinite) periods of time?

*And how come a neutron has no charge since and a proton does? They are both made of the same type of quarks and they both have no movement.
 A: 
1    How does an electron get its charge?

This is the elementary particle table . The electron is an elementary particle and its charge is an observable attribute that , together with its other quantum numbers and mass, classify it as an electron. 



  
*And how can it maintain that charge for very long (infinite) periods of time?
  

Observations gathered over a century have not shown the decay of an electron, i.e. of losing charge and thus becoming another particle. So it is by construction of Nature.


  
*And how come a neutron has no charge since and a proton does? They are both made of the same type of quarks and they both have no movement.
  

Look at the quarks on the table. The exact quark content has to be added up, and the charge added. 
Proton is up+ up +down =+1 , and neutron is down+ down +up  =0.
So the general answer to

How do particles get their charge?

is, it depends on the particles, if they are elementary or composite. Composite one get their charge by the addition of the charges of the elementary ones they  are made out of. Elementary particles have been defined by the study of the results of innumerable experiments, over more than a century. A minimal mathematical model called the standard model of particle physics assigns them as a basis for describing the underlying quantum mechanical level of nature. This model has been very successful in describing all known interactions and predicting new observations.
A: An electron is a fundamental particle(Lepton) and is different from both protons and neutrons, which are not fundamental(i.e they are made from even more smaller particles called quarks). For an electron, the charge it has is an intrinsic property, that is it is a part of its description along with mass and spin. 
Coming to neutrons and protons, even though they are made up of same types of quarks, the neutron is made up of 1 up quark(charge=2/3) and 2 down quark(charge=-1/3). So in total its electrically neutral. For protons, they are made of 2 up quark and 1 down quark making it carry 1 unit of charge.
A: I would answer this question differently. From other perspective, a electron gets its charge by the only generator that is not broken after the S.S.B of the SU(2)xU(1) gauge group. In this case $$ Q = \frac{1}{2} Y + T_{3} $$
Where $ Y $ is the hypercharge eingevalue and $ T_{3} $ is the eigenvalue related to the SU(2) diagonal generator. So, as every elementary particle as a value for T_{3} and $Y$, we can obtain their charge by this combination.
For example, the (left-handed) electron has Y=-1 and $T_{3} = -\frac{1}{2}$, therefore we get a Q= -1. 
Concluding, we can say that "the electric charge came from the combination of the generators' eigenvalues."
P.S. In the SU(5) gauge group we even get the quantization of the charge as $ \frac{n}{3}e$. Where "e" is the fundamental charge we already know.
