I've seen a few questions on this, so I wanted to get a topic setup for discussing weapon upgrades, and the general formula I've used as far as upgrading.  I'll list a few basics, and my general methodology, though other people are welcome to offer their own suggestions. 
 
Basics: 
Weapon upgrades is a concept available in Chapter 4, though generally very inefficient to do at chapter 4.  Chapter 9-11 is when it finally starts to become much more viable.  It's an expensive and time-consumming process that can be boiled down to a few core concepts.  Basically, every item in the game has levels and XP, you increase the parameters of the item by levelling it up via putting components into it.    Eventually your item will reach a max level and cannot be upgraded any more.  In the case of weapons, you can then use a specific mineral item to upgrade the weapon into an entirely new weapon, and continue upgrading it the traditional way.  Each weapon can be upgraded with minerals twice.  With the 8 basic weapon types for each character, this means there are 48 weapons in the game with 3 tiers each, or a grand total of 144 weapons.  Every character's "tier 3" is considered the same ultimate weapon however, with the same name and appearance (though the stats are based on what you levelled it from). 
 
There are two basic types of components: 
Organic Components: These are purchased from "Creature Comforts" and dropped off organic monsters, basically these are monster parts.  Organic components are low cost and give very little XP to an item.  However, they build up a multiplier that makes future upgrades give more XP to the item.  This multipler caps at 3x. 
 
Mechanical Components: These are purchased from "Lenora's Garage" and dropped off mechanical enemies.  Mechanical parts are high gil cost and give a lot of XP to an item, but will rapidly remove the XP bonus gained from organic parts.
 
Efficient Upgrading: 
Basically the idea is to use organic parts to increase a weapon (or accessories) XP multiplier to x3, and then to use mechanical parts to cash in on the XP bonus and level up the item.  Ideally, you will want to level the item to the level cap in one go (because your XP bonus will be gone after using lots of mechanical parts).  Here is where some math comes in, figure out how much XP you get for an item at 3x, and then figure out how much XP you need to reach level cap.  You don't want to waste excess parts by putting too many in, but you don't want to undershoot and waste your bonus. 
 
Generally speaking it varies, but it takes roughly 50k XP for a weapon to reach it's first level cap, 450k XP for a weapon to reach it's second cap, and 1.5 Million to max out a weapon totally.  That's a lot of XP, so you'll want to figure it out beforehand to save parts.  Aim for those numbers and you should be pretty close.  
 
In order to save gil, I sell most any mechanical part that isn't very efficient, because mechanical components are worth more gil and you can make a fair amount of money selling them.  I rarely sell organic parts, as they just aren't worth much and all are about roughly equally efficient.
 
Parts to Use: 
Organics:  There are quite a few that work 
Sturdy Bone, Barbed Tail, Vibrant Ooze (the infamous Japanese colored jelly) x36 = 2,880 gil and will bring a weapon straight from 0 multiplier to x3. 
Otherwordly Bone, Diabolic Tail, Transparent Ooze x24 = 3,600 and will also bring a weapon to x3.  These are the second most efficient. 
You can use any combination of these for the same result as well. 
 
Mechanical: Less efficient ones, and more expensive so be more careful here.   
Ultracompact Reactor: 40,000 XP and 50,000 gil from R&D store (The most efficient item to use in the game) 
Supercharger: 770 XP and 1,600 gil (you'll probably use these a fair bit) 
Particle Accelerator:  4,800 XP and 10,000 gil from R&D store (another great one to use, pretty efficient and a lot cheaper than ultracompacts) 
Superconductor 400 XP  840gil     (You'll get these the earliest I believe) 
 
Crankshafts are also a good item available earlier than any of the above, if you want to start upgrading.  Generally, I sell any mechanical item that isn't one of the above, to make gil in order to buy these items.  
 
Important Stores: 
Creature Comforts: Unlocked during the story.  Sells a large number of organic parts. 
Lenora's Garage: Unlocked during the story.  Sells a large number of mechanical parts. 
The Motherlode:  Unlocked during the story.  Sells minerals that are used for upgrading weapon tiers. 
R&D Facility: Unlocked from Cie'th Stone Mission 7.  Sells the higher end upgrade items (like reactors)
 
Anyone else have any suggestions or tricks they found while weapon upgrading?  Most of what I posted was already common knowledge from the Japanese release, just clarified and combined in one place.

Dude...fight the giant turtles, use Vanilla's Death + Summon.  You get two items.
 
One sells for 150k, the other sells for a lot less and is required to evolve ultimate weapons.
 
Buy 99 Jelly for 150 gil, and then buy as many Superconductors as you can.  Use 36 jelly on any item to get the 3x exp boost and start using those super conductors.
 
We found out within a week of the Japanese release that this is all you need to do to max out all weapons and items for cheap and quickily.
 
EDIT:  Make sure you have Good Choice and Best Choice or else be prepared to fight a lot of turtles, you should always have those two items and you should also have CPX2 accessory (Growth Egg) so you get 80,000 CP from the turtle.

Great post! When would you recommend upgrading then? I'm currently on Chapter 10; I've upgraded a little bit here and there but since finding some really efficient upgrading methods on the internet I've kinda stopped, plus I didn't really think it's necessary at this point. Asides from the occasional boss battle I haven't had much trouble so far but I have done a bit of grinding here and there. I have a fair amount of gil (literally haven't spent any of it at all yet) probably just over 150k and I'm sure there's a few things I could sell as well. Should I wait until I've beaten the game to start properly upgrading or what?

this helps, but one thing i dont get it, you upgrade weapons...and then you find new ones (with different passive abilities) that are weaker, so you have to upgrade THOSE if you want those passive abilities, which kinda makes upgrading the old ones pointless? or am i missing something?

" this helps, but one thing i dont get it, you upgrade weapons...and then you find new ones (with different passive abilities) that are weaker, so you have to upgrade THOSE if you want those passive abilities, which kinda makes upgrading the old ones pointless? or am i missing something? "

" this helps, but one thing i dont get it, you upgrade weapons...and then you find new ones (with different passive abilities) that are weaker, so you have to upgrade THOSE if you want those passive abilities, which kinda makes upgrading the old ones pointless? or am i missing something? "

" Dude...fight the giant turtles, use Vanilla's Death + Summon.  You get two items.  One sells for 150k, the other sells for a lot less and is required to evolve ultimate weapons.  Buy 99 Jelly for 150 gil, and then buy as many Superconductors as you can.  Use 36 jelly on any item to get the 3x exp boost and start using those super conductors.  We found out within a week of the Japanese release that this is all you need to do to max out all weapons and items for cheap and quickily.  EDIT:  Make sure you have Good Choice and Best Choice or else be prepared to fight a lot of turtles, you should always have those two items and you should also have CPX2 accessory (Growth Egg) so you get 80,000 CP from the turtle. "

" @Azteris: Uh, Good Choice is in item shop...forget which one, Best Choice is from the evolution of Good Choice.  Growth Egg is from Mission 36. "

I think they are called 'Collector Catalog' and 'Connoisseur Catalog' in the English version rather than Good Choice and Best Choice

I read somewhere that lightning has both a Ultima and Omega weapon anyone known this to be true

http://finalfantasy-xiii.net/forums/showthread.php?t=10251

found it here

What's the catalyst used to upgrade Good Choice/Collector's Catalog to Best Choice?
 
EDIT: Mnar Stone for those also wondering.

Thanks for setting this up, bub. I'm at the point in the game where Weapon Upgrades can be done. I'll wait until I'm at the recommended Chapters before I start tinkering with my weapons.

You get the Growth Egg from mission 55, not 36.

By "supercharger" you mean "Perfect Conductor" I think?

A couple of questions
 
1) When do these 'superchargers' or 'perfect conductors' become available in the shop? I'm at the begining of Chapter 11 and can't find them
 
2) Does there exist a list or a chart of the weapon exp requirements and such? I've already upgraded most of my weapons to level 8 or so but this was before I realised it would be better to upgrade each weapon to tier 2 in one go to make better use of components and gil. I've already got them all up to 3x multipliers, now I need to know how much mechanical components to use on Gladius without wasting any.

great topic bump

" @BomberDude: Save it for Chapter 11 and after you beat the boss on the bridge.  Warp back to Grand Pulse and learn death and farm the turtles...it's the quickest way to make cash.  Do Missions until 20 or as much as you can do.    Like it states on Japanese and import forums...buy 99 Jelly and then buy Super conductors.  You MAX out all items and weapons in seconds. "

As a question is spending a lot of effort trying to get the most out of upgrading weapons/accessories in the early parts of the game necessary or worth it? I just unlocked upgrading in chapter 4 and am sort of just putting in organic components to raised the multiplier, just throwing anything in an item really. Then using tech components. From what people say on boards and the guides, it really comes into play in chapter 11 when you start grinding a lot more. Is that how it is?
 
I guess I'm gonna spend a while, I'd like to S rank the game since I plan on stretching ff13 out as much as I can really so if it helps in the long run I'll plan out my upgrading better.