Hey, I now have enough points to buy a new champion. I've been using Mordekaiser for a little while and I love playing him, but I feel he's lacking when it comes to really helping the team.

I've been reading up about various tanks and the ones that stick out most for me are Rammus and Taric.

I'm curious which of these would be a good choice to try out for a tank role. I'm mainly wanting a champion that doesn't die easily and can support his team effectively, as well as push during early game. Taric, from what I hear, fills that role. BUT, I haven't a clue how to build him or anything like that.

Any tips? Also, between the two, which would probably be easier to get into? I'm still kind of new to the game.

I think Taric would be easier for you to get into. Just because you can keep healing yourself and your teammate. If I would really suggest you an easy tank, I will recomment alistar with Flash. He is fairly easy to play and very hard for enemy team to kill him because of his ultimate.

Rammus is kind of hard to play in low level. You need to last hit occasionally and you need to keep up leveling. Once you get your ultimate, you are quite devastating.

Taric is more versatile and his ultimate is perhaps the best team ultimate out there. However, he isn't a full fledged tank. He is better suited with another strong or meaty character. Rammus is a solid tank choice if you want to be the main meat, but if you really want some fun and to be REALLY versatile, try Shen.

According to your wants, Taric would be better. Both Rammus and Taric are very hard to kill, but Taric provides better support than Rammus and pushes better than Rammus in the early game. For what it's worth though, Rammus is better at defending against pushes.
As for ease of use, I'd say Taric is easier. His skills are very straightforward: heal whoever's low, stun whoever's farthest, shatter in the middle of everyone, turn on ult for teamfights. Rammus' Powerball has some odd hit detection, and using it cancels the use of some other skills (and vice-versa).
In the end, Rammus is the truest tank in the game with his Taunt and DBC. Taric is not as good at taking damage but is more versatile. Imo you should just wait until they're free and try before you buy.