Still feeling lost in this game regarding achievable goals to work toward in the beginning, but I'm doing a bit better. I saved up enough Jewels to pull once and here is what I got:
6 star Owl
6 star Rock Titan
5 star Xemnas
5 star KH CoM Black Coat Riku
5 star Xigbar
5 star Ice Titan
5 star Xaldin
5 star Timeless River Donald
5 star Vexan
5 star Ansem the Wise
and I just got two 5 star Classic Donalds from the event.
Who should I be leveling up and using? I don't know much about upgrading with Cid/Chip or whatever. Right now my plan is to just level those to their caps and keep going on with the quests. Anything else I should be aware of as I progress?
Owl is pretty bad (though if you don't have any other medals with similar function, you might find use for it here and there) and Vexen might be the worst Org XIII medal unless I'm mixing him with someone else, but otherwise that is a pretty good pull. Ansem the Wise is pretty worthless since it only raises attack for its own special attack and the attack following it. But keep it around for leveling up Nova once you get to the point where you can guilt medals.
Xemnas is probably one of the strongest single target (power) medals in the game so you can't go wrong with leveling him up. Xaldin is a really strong medal as well. BC CoM Riku is a decent AOE magic medal and Timeless River Donald is a good magic single target medal. Both probably good medals for you still, so as far as magic medals, focus on those.
Rock & Ice Titan are a bit costly as far as Special Gauge usage goes, but they are really strong for AOE medals, so they are worth keeping in mind if you have some quest where you have difficulty defeating some group of enemies.
Xigbar is a really good medal once you guilt it. Before that it's not necessarily something you'll want to have in regular usage (unless you of course don't have anything better) and even after guilting he's best in quests that have (groups of) enemies that are of different attributes, since he has an attributeless special attack (which means that he doesn't get the extra-multiplier against Magic-enemies that Speed medals are normally strong against, but neither does it become weaker against Power enemies that it's weak against).
In short, these are worth focusing ASAP:
Xemnas
Black Coat Riku
Timeless River Donald
Xaldin
Xigbar
Owl, Vexen and Ansem the Wise are whatever, the Titans are strong but not medals you'll have in regular rotation.
Cids are only for use with 6* medals. They bring the Cost of (equipping) the medal down, which is important since 6* medals cost a lot to equip without any Cids spent (46 or 47). You can't have a keyblade with all 6* medals equipped if you don't spend Cids on them, even when you are leveled up enough to have Max Cost for your character. You can only use 10 Cids per medal, so that's -10 to their cost, at most.
Chips are ADDITIONAL Strength for any medal. The max amount STR you can give to medals with Chips is limited by the medals *-level. A 6* medal can have an additional +1000 STR given to it with Chips.
At the moment just focus on playing through story quests and leveling up the medals you have and check out the Events quests and do everything you can in those at the moment and just let everything else come as your understanding of the mechanics & medal system grows.