"imo this doesn't sound like too herculean of a task"
... in many ways I reckon I would have an easier time teaching graphics, text or level editing from scratch.
Still it is a doable thing.
As above if there are cheats available, and it is a somewhat popular class of cheat, then go for them, also grab them (and confirm they work) as they will be helpful in the next step (not that finding a cheat is usually all that hard).
There are three broad approaches to exp and level up mechanics.
1) Premade table of values at which level up happens, and what monsters/actions (there are some level up by action games on the SNES, and many exist on later setups) give you.
2) Complicated equation governing things. If you have ever played something where fighting low level enemies gives you 1xp where at the start of the game it was more like 100xp then this is that in action. Can also be something like next level = current level multiplied by..., which possibly has some consequences if you edit it and use the save with a stock game.
3) Random. For whatever reason I mostly see this in strategy games (Shining Force being among the more annoying for this one for me/reason I learned to love savestates), though pokemon has elements of it as well.
Can exist in all but you sometimes have effects that boost exp -- attack from behind, first time back after a break (usually done in MMORPGs to stop people rinsing the content by giving them a reason to have a break but have seen them in more conventional games, don't know for the SNES though), don't take damage... If you can twiddle such a flag in a cheat for the perk then can be worth a look.
You can also bring the mountain to Mohamed, which is what a lot of things go in for rather than messing with tables.
Usually at the end of battle your exp is totalled up and it is added to the exp pool (this would also be where the effects thing above might be present), same for gold, other thing that is technically also experience, dropped items...
Anyway between/during this summation/calculation and the addition to the pool you sneak in a nice little logical shift in whatever direction you want for your hack (some people like 0.5x exp hard mode/grind mode) as logical shifts are the binary equivalent of multiplying by 10 in decimal in that you shift decimal points as it were around, though being binary the progression is 2x,4x,8x,16x,32x... as you might well have seen in such things in the past. Technically you can do other things -- 3x = 4x-x or 2x+x, or you can arbitrarily add or subtract from totals if it balances the part you want (might also solve overflow bugs) but cross that bridge as and when.
If there is an in game item that does it for you then figure out either how to give that to your party, or possibly just its effect (things like no battles potions/charms tend to be a flag in the game somewhere just like being poisoned or paralysed or whatever status effect, albeit maybe in a different location. Same deal for this, so you do a cheat search equipping and unequipping the item to find where it changes).
Beyond that you have two main approaches I am going to suggest (there are some further ones in fiddling with vblanks but they are dubious in both quality and safety for your hardware).
You change the additions used in the movement. Most games will have something of an internal map of where you are in a level/screen (single screen might even use the screen layout itself but different discussion there). Rather than add one you add two, though quite often you will have made a walk through walls cheat if you do this.
You alter the animations to skip steps, be quicker or otherwise take less time. For text if the game has some kind of speed option then find the corresponding flag/value and see what putting other things in gets you -- if it is straight speed then great as you might have higher values available, if it is more of a value thing then there might be a hidden value used at points in the story but normally it only allows some things, or it might do nothing. I tend not to see that for graphics outside of things that give you the option to disable them, though you might be able to turn this on and off according to your whims rather than having to go all the way back to a menu.