Protips from me (though I'd hardly call myself a seasoned vet):
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Accessories are better than any weapon you get initially
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Especially if you plan on Grinding. You get the same amount of EXP regardless of what level you're fixed at (even in Fixed Level Gateways in order to get KP bonuses, reducing your level doesn't give you more EXP it just increases the difficulty and gives you more KP). Reducing your level limits what kind of equips you can use anyway, so there's no point in focusing on that until you can make some items with special effects built into them.
The main focus of grinding is to grab Bonus EXP battles and maximize their grind potential.
I always stick with Hardcore/Fat Chocobo/Blue Chocobo course Play Plans. Fat Chocobo keeps giving you Chocobo Wings, these are useful later on in Scenario 000 (Gateway of Fallen Snow) where you can use them an unlimited amount of times without worrying about the item breaking, so long as you chain-battle on the "Accessories Protected" panel. Once you get "Diamond Ring/Stud/etc and Dazzling Diamond/Beckoning Cat" Accessories, you can further grind your party and get 56+ AP in a single fight or more depending on your setup, without worrying about breakage. Having Hero Essences/Hero CP+ items also speeds up the process too.
Then you can equip stuff like Arcane Resin/Brave Resin and go nuts while grinding AP/EXP/Gil all day in that Gate. Besides the fact your party members can get 11+ levels a pop for tough enemies on good bonus battles, you get a ton of AP/PP/Gil/KP besides if you do it right.
And once you start rolling in KP you can start making the items you REALLY want...that includes ultimates.
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Learn the Item refinement process
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This should be obvious, but if there's an item you haven't made that's not in your inventory, chances are you should buy/make it. It unlocks more items in the shop.
Any items you come across that you don't typically get sometimes unlock items you can buy/create too.
Having a surplus of items is also a good way to boost your Gil gain on Bonus Day, which can be manipulated by the calendar.
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Evade is better than Jump
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When fighting enemies, enhance your Evade abilities as best you can, evade distance and evade frames help you dodge stuff like Sephiroth's Hell's Gate that normally wouldn't be easy to dodge.
Heck, I equip "Have not yet Jumped" as a Booster Accessory since I rely on Air Dash/Free Air Dash more.
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Coward enemies are stupid enemies
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Most enemies that try to dodge everything can't dodge more than ONE thing at a time, so throwing 2 HP attacks at them in a corner at the same time will usually clobber them.
Use Assist HP attacks, use multiple projectiles, trap bait, whatever it takes.
Taking advantage of Chaos Judgement is also good too, since Chaos will get angry and FREEZE them for running away.
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KP Switch + Multichain + KP Gambler
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No seriously, this combination will net you any amount of KP you desire, and all you have to do is WIN.
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READ BATTLE TIPS
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Selphie's advice is solid advice, I kid you not. Manikins that love to block and counter your attacks can easily get thrown off if you take just a MOMENT to pause before attacking.
Throwing a "relatively safe" HP attack at someone to corner them is also a good idea.
Precision Blocking pwns a lot of things that normally would hit (like Zidane's Grand Lethal)...you should try it.
Many HP attacks cut through "projectiles" if you time it right. Cloud's Braver can cut through Garland's Tornadoes easily.
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Make your own rules
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I'm not kidding, this is key for setting up certain conditions. Stuff like extended EX Mode or Assist boosting will make your life that much easier in some fights.
Maximizing Warp Trap Damage also helps too. Sure, it hurts, but you're smart enough not to get knocked in, right?
Also, any summon/item you find can be BANNED. Keep this in mind. It works on enemies a lot.
Bravery Bonus can be reversed to give characters with Lower Levels 9999 Brave, but this only works in Quick Battles.
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Gabranth is your friend
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So long as you use him for grinding, the most he can do is break you. Even at Level 100.
If you play your rules right, you can set up quick grinds and speed up the process.