General advice: get familiar with Processing first. Look at the examples shipped with it, particularly the Topics > Motion section, which has several sketches with moving balls.
You probably need to learn the rudiments of object oriented programming (there is a Learning section on Classes / Objects) to reach your objective. Perhaps only one ball class is needed, it seems the other kind just has a null speed and are not attracted by other balls.

Thank you for the advice. I've definitely been looking at those. The problem I'm having is that I've figured out how to make a bunch of moving balls, and I've figured out how to make two balls stick (one of them being controlled by the mouse), but I can't really figure out the combination of the two. Through my limited experience thus far, I've found that I struggle with applying multiple concepts.

I think what phi.lho was pointing out was not to think of them as two different kinds of balls. Use one kind of ball and just set the speed to be 0 for the stationary ones.

Hello,

yes, make yourself comfortable with these examples

bounce at walls (you need this too)
http://www.processing.org/learning/topics/bounce.html

multiple balls:
http://www.processing.org/learning/topics/arraylistclass.html

two balls collide:
http://www.processing.org/learning/topics/circlecollision.html

When you are really comfortable with these examples - and only then - try setup your code and post it here.

Then we can help and then you have the level to do it.

Otherweise your plan is too big to accomplish (with all due respect).

BTW. when you want to stick two balls together and move together just store their current position (nearly the same) and move vector (same).

Greetings, Chrisir