How can I show my significant other that I care about them?

Go out of your way to do little things that don't directly say that you care, but imply it. Unless your significant other is completely oblivious he/she will pick up on it pretty quick. Little things like buying food you know they like, or getting yourself involved in something they feel passionate about (sport, gaming). All the best!

I am a quiet person as well. I think the best way to show that you care is asking keeping up to date with them, so to speak. If they like sport, a certain tv show, etc., ask them about the scores of the last game or if a new episode came out. If they participate in an activity, ask them about that, too. I find texting nice compliments is easier than saying them in person, but make sure you say nice things every now and then so you don't look like you're "faking it". Praise them when good things happen, comfort them when they are sad, remember important dates, remember the things they like, remember things you do together, anything that you feel is a nice thing to do that's in your comfort zone.