I knew a few things about soldering, and I know a few more things now.
The first capacitor I replaced took about 30 minutes for me to do, in order to figure everything out and get the process down. I knew basic stuff like, heat up the metal (solder), etc. and I pretty much just figured it out. Now it takes 5 or 10 minutes to do a capacitor. I knew what iron to buy (a 15 watt one with a smaller tip, which is more suitable for these things) and some stuff like that.
Solder does not seem to affix to plastic. There is no metal left on the board for this particular capacitor except for what's inside of the little holes. Any solder I put there falls off because it does not affix to anything.
The pads broke because I removed the capacitor too aggressively. I cannot seem to remove them by simply heating up the pads because most of the pad is under this plastic stuff, so I have been rocking the capacitor back and forth gently until it breaks off, and then I remove the little metal bits it left behind from the little globs of solder.