I think I'm in a near-similar spot to where I'm My Own Milf is in Chapter 9; that's the best first place to really start working on upgrades, seeing as you can get some stuff maxed out real easy.
I'm probably going to start working on an FAQ for the equipment upgrade stuff, between garnering what information I acquired from the official strategy guide (I couldn't resist that hardbacked beauty) and my own observations. But, to at least give those in the dark some clue as to what's going on, here's what I can tell you off the top of my head.
Basically, there are two types of components: organic and non-organic. Organic components ALWAYS work to increase your multiplier bonus, but provide little XP themselves, whereas the non-organic components ALWAYS work to decrease the multiplier, but are worth substantially more XP.
To get the most of it all, you always want to start by maxing out the multiplier with organic goods (Sturdy Bones are the best to buy from the Creature Comforts shop initially). Then, you'll want a huge stack of some kind of non-organic component (I don't remember which is best at first, but I'm currently buying Crankshafts from Lenora's Garage, which yields the most XP per gil out of the selection) to apply to it all at once. The bonus gets recalculated after putting in a stack of items, so just try to keep it all to one type of component for best results.
In terms of the XP required, there's a base amount required per level (the amount at level 1), and there's an amount that's added each level to that base amount. For example, say you have something that starts at Level 1 and maxes out at 6. On the first level, it takes 100xp to get to level 2. Once you level, you notice that you require 120xp to get to level 3. Your "step" amount is that 20 that was tacked onto the next level, and that 20 gets added each level, so 3 to 4 requires 140xp, 4 to 5 requires 160xp, and 5 to 6 requires 180xp. Add those up (100+120+140+160+180), and you see that the item requires 700xp to go from start to finish.
If you know how much XP you'll need to get to max, I suggest just doing it in one fell swoop; just know how much to put in, so you don't over-do it (the game will let you put in as many of something as you want at once, regardless of whether or not it'll max out). Unfortunately, there's no way to know the max level of an item in-game without getting there (or buying a book that has all the item max levels listed, LOL), but I do know that all the lower-rank weapons (the ones that are in shops) max out at either 21 or 26, to give you something to start with.
Once you max out an item, if it can be changed to a higher-ranking item, you'll need the appropriate catalyst component. Each item has only one catalyst attached to it to get to the next rank item (and it won't let you use it if it's the wrong one, so no worries about wasting items to find out, LOL). Now, the downside... Once you transform something to a new item, the previous XP goes away, and the new item is back to level 1, so be wary of that (ESPECIALLY with weapons, which you'll want to prepare to re-level immediately).
As MILF pointed out, I definitely liked Chapter 9 as a good starting-point to really grind stuff out; a lot of the enemies there drop Credit Chips and Incentive Chips (which are only good for selling, and worth 500 gil and 2500 gil respectively).
In response to Shlup, all the weapons you find are the "starter" rank weapons. Each character gets access to 8 different weapon "types", and they all have different stats and abilities to them (one may have higher strength but sacrifice magic, and another may be lower in both strength and magic, but make you faster, for example). Each of the 8 weapons has three ranks, but you must max each rank to upgrade to the next.
Kuja Highwind/Blackmage: Some guy that used to be active back around 2001... He almost made his way through Proto's Custom Title Tournament way back in the day...but almost only counts in horseshoes and grenades. Oh well.