Don't worry, there are enough Quickening panels so all characters can gain all three quickening that they have. Also, the second and third Quickenings you gain will double and triple your MP just for acquiring them.

Quickenings consume a chunk of your MP and once you start using Quickening chains it will begin blowing all your MP. Your party is allowed three Quickening each and each is stronger than the last but consumes more MP. When two party members gain Quickening's you can start using Quickening chains, basically you have one activate their Quickening and during the animation for it, you'll see a list in the right bottom corner listing applicable party members (they have to be in your active party) which will either list three options next to their names:

Grayed out Quickening name = Can't use a Quickening
White colored Quickening Name = Can use that particular Quickening
Mist Charge = Recharges all their MP and allows them to perform a Quickening on the next turn.

A timer is also on the bottom showing how much time you have to input commands. You can press R2 (I might be wrong, its been awhile) and it will shuffle your Quickening options and so you basically shuffle until you either get a highlighted Quickening name or Mist charge then you activate them.

What this all does is allow all your party members to chain together their Quickenings, think of it like this:

Cloud uses Omnislash and during the attack, the game gives you the ability to let Tifa activate her Limit Break immediately after with no break for the enemy, so you chain them together; but while Tifa's is doing her limit break the game gives you that ability to recharge Cloud's Limit gauge so he can use another Limit Break after Tifa finishes. Of course their is a timer going on that prevents you from doing this indefinitely. Also, even if Barret your third member has a full Limit Gauge, he can't use it until you get a shuffle that will allow him to use his and many times after everyone goes, you may run out of time shuffling through the options to get a recharge.

Now once you start acquiring level two Quickenings, your Quickening chains can start making use of the Concurrence system. As I mentioned earlier, all Quickenings have three levels of power and that is actually important when you are making Quick Chains cause you can activate special damage bonuses by making Quick Chains consisting of specific Quickenings of a particular level. Course you don't have too much control over this but luckily the game doesn't need you to go into any specific order and it will always do the most powerful Concurrence you are qualified for. So really in the end, the game gives you even bigger damage bonuses for just keeping the Quick Chain going.

This may all sound very confusing right now but believe it will make perfect sense once you start using it and its actually much simpler than I probably explained. Of anything, unlock Quickenings so you can get the MP bonus.