First off Snow, Vanille, and Sazh are all garbage. Get them out of your party now. They offer little in comparison to Lightning, Fang, and Hope and I'll tell you why.
Lightning
She has a higher balance (like most main characters in FF) of Strength, Magic, and Toughness(HP) and also gets Raise quite early which makes her a critical element in your party. Having Commando, Ravager, and Medic and the highest ATB (for a while, before the rest of the characters catch up) elects her for the Party Leader. 90% of the time you'll find Lightning as a Ravager because building skill-chain is very important and I'll go over how this works with the paradigms that I'll setup shortly. Overall Lightning is the most balanced character and can really save any situation (I've been down to critical health on with Lightning as the only one left alive and have restored the whole party to full hp and WON) <--- incredible moment
Fang
She's got the most strength (maybe not when you get her but her str gets higher than any other character) which makes her the best possible character for a Commando class (which can sometimes be extremely important expecially when you max the skill chain bar). But, alas, you won't find Fang acting as a commando as often as Sentinel, which would make you think hmmmmmmm wouldn't Snow be better because he has more HP? The answer is no and this is why. Fang, when acting as a sentinel, won't take enough damage to kill her 90% of the time and the fact that she has a little less health actually helps out when trying to restore full party HP. If it takes 3 cures longer to heal Snow, that could mean the boss your fighting has THAT much longer to kill you. Fang also has saboteur which can help out later on with Pain and Fog and keeping you from losing the Skill Chain bar when buffing/debuffing. Overall Fang is around to hog damage and keep the skill chain bar from dying (which is 90% of the combat later on) with the occasional debuff.
Hope
I used to hate Hope early in the story because of his abscense of testicles but later on he gets incredibly strong and has enough HP to withstand most big attacks. What really sets hope apart is his Magic, he gets really strong (like Fang and her massive Strength) and lays down some heavy attacks. I found that leveling Hope's Medic ability before even touching his other Tier 8 circles really helps when he gains Raise. (You'll see why when we setup the paradigms). Hope also has Synergist which completes the party in order to have every job possible. There's not really alot to Hope, he spams magic and heals and on big fights has incredible buffs to prevent all sorts of damage and imbue weapons with higher damage (elemental and later on Bravery).
Paradigms
The one thing you want to focus on with Paradigms over ANYTHING else is maintaining skill chains. Sometimes putting skill chains before healing can be fortuitous, but that element is left up to your judgement in combat.
Paradigms: (Lightning | Fang | Hope)
Relentless Assault: Ravager + Commando + Ravager
Delta Attack: Commando + Sentinel + Ravager
Combat Clinic: Medic + Sentinel + Medic
Solidarity: Commando + Sentinel + Medic
Mystic Tower: Ravager + Sentinel + Ravager
Evened Odds: Medic + Saboteur + Synergist
By starting with Delta Attack you get the most systematically effective attack. Delta Attack is the foothold for the start of pretty much any battle (or enemies that you haven't started building the skill chain on). Fang will start soaking up the damage while Light and Hope start building the skill chain bar.
Following Delta Attack, Solidarity is the best viable option when Fang (or other members) lose health. It keep the skill chain bar from dropping while continuing to inflict damage and restore health. I've found leaving Solidarity up until everyone has full health or very near full health is the most effective.
After solidarity, Mystic Tower is the best option 99% of the time and you will probably want to switch between solidarity and mystic tower until the enemy is staggered or close to that.
Once staggered, Fang has the ability to launch most enemies and Relentless Assault will ensure maximum damage potential. (If unable to launch, Mystic Tower can sometimes be a better resource.
In most fights, as you may have noticed, you will want to keep Fang as a Sentinel until it's clearly safe to drop into Relentless Assault. Otherwise you will find yourself getting mercilessly slain when trying to regain hate (as all hate drops when a Sentinel changes roles). Also as another note, I didn't talk much about Combat Clinic as it's not a main part of combat and should never be left up for longer than two full ATB charges unless A) Hope is dead (where you should raise him immediately if Lightning is anywhere above critical instead of using auto heal) or

The enemy hits you so hard you can't recover HP fast enough to keep the skill chain going before another big hit.