I've heard there is some "cheese" that would allow you to enter mystic theurge without going wizard 3 / cleric 3: how do you do that? What is the earliest you can enter the class?
You can enter the class as early as 2nd-level, but doing so is pointless as you have no second class to advance. (Arguably, this could advance one class at double-speed, but that’s a terrible idea even for a high-optimization game.) Thus 3rd is the earliest meaningful level to do it.
This is pretty typical early-entry stuff; there’s a whole handbook on that. I tend to find it pretty hard to read, but it is incredibly thorough.
There are two issues for entry this low:
No class can cast 2nd-level spells so soon without shenanigans
Your maximum skill rank at 2nd level is 5, and mystic theurge requires 6 ranks in two skills
The following constitutes some fairly high-end optimization tricks. In the case of a mystic theurge, using some tricks is almost necessary, at least if you are to keep up with your single-classed brethren, but these tricks can be used in other cases where they just lead to overpowered characters. For example, easing your way into mystic theurge is one thing, but doing the same to enter dweormerkeeper is game-breaking.
Even in the case of the mystic theurge, though, I recommend that you talk with your DM rather than use these, even if they are not necessarily game-breaking in that case. The mystic theurge is a very weak class, and most, I think, recognize that. If your DM does, or you can convince him, it is far better for the game overall that you work out some alternate entry requirements for that class, rather than set precedent for the use of these tricks. If your DM cannot be convinced of the weakness of the mystic theurge, using one of these tricks is very unlikely to garner you much respect; most likely they’ll just be banned and the DM will be angry at you for trying to slip something like this in.
There are many ways to get the ability to cast higher-level spells than you should be able to; in most cases these are very limited, or only “effectively” higher level, but they’re good enough for prerequisites, including the mystic theurge.
Precocious Apprentice (Complete Arcane pg. 181) is the easiest, being only a single feat. That gives you 2nd-level spells in any one arcane class. Some argue it needs to be combined with something else to allow you to cast spells as Precocious Apprentice gives you literally one single 2nd-level spell slot which can only cast one spell; personally I think that’s a pretty weak argument (sorcerers only learn a single spell of a given level when they first access it, and I don’t think anyone is barring them). But even so, there are ways to get around that.
That only covers one side, however. It does mean you can do Wizard 1/Cleric 3/Mystic Theurge 10 and only lose 1 level of cleric for 11 levels of wizard, which is solid, but your wizard spellcasting is behind.
The easiest (read: works without relying on variant rules like flaws or elder evils, or on DCFS and similar high-cheese) way to get both is to go with the sha’ir from Dragon Compendium as one of your classes. The sha’ir, weirdly enough, casts both arcane and divine spells. Thus, when you use Precocious Apprentice for sha’ir, you can arguably pick a spell that is available to you as both arcane and divine (Precocious Apprentice requires an arcane casting level, but its benefits don’t mention anything about arcane vs. divine), and your single 2nd-level spell per day can be castable as an arcane spell or as a divine spell, at your option. If that’s a no-go, Alternative Source Spell from Dragon vol. 325 explicitly lets you treat an arcane spell as a divine spell or vice-versa, though it’s got a weird statement about power source that Customer Service read as not allowing you to qualify for mystic theurge. So if that’s out, you can try Mad Faith from Heroes of Horror; if you have moderate depravity, you get a 2nd-level divine spell slot.
So a human (for the bonus feat) sha’ir 1 with Mad Faith and Precocious Apprentice can cast a 2nd-level arcane spell and a 2nd-level divine spell. Take your second level as any arcane or divine spellcasting class, and then your third level can be mystic theurge, advancing that class as arcane or divine as appropriate, and sha’ir as the other since it can benefit from either.
Replacing sha’ir requires a source of additional feats. There are numerous of those, but they get ridiculous fast. And of course, there’s always Pun-pun: it’s possible for a 1st-level character to get all of everything, which is more than sufficient to qualify for mystic theurge (or any other prestige class).
Skills are trickier, sort of. The typical approach is to acquire the services of a high-level bard and mid-level psion, though there are tricks to avoid the psion (and there may be replacements for the bard). The approach is to have the bard use Inspire Greatness on you, which gives you two more HD than you should have. Your skill rank maximum is now 2 higher than it would otherwise be.
However, Inspire Greatness doesn’t give you an opportunity to gain or assign skill points, which is where the psion comes in. The psion can use psychic reformation to allow you to rearrange your skill points. If you do this while under the effect of Inspire Greatness, you can rearrange your skill points such that you have the necessary 6+ ranks in Knowledges (arcana) and (religion).
The psion can be replaced by anything that lets you assign new skill points (and enough of them to get the ranks you need). Tricks to get new feats could get you Open Mind for 5 skill points. If the bard is undead or warforged or something that doesn’t need sleep, it could just continue playing Inspire Greatness through your level-up from 1st to 2nd. Other possibilities exist.
Then, once psychic reformation (or whatever) is complete, the bard can stop with Inspire Greatness; you will keep your skill ranks where they are. When you level up again, you qualify for mystic theurge and can choose it as your 3rd-level class.
There is an infamous feat in Complete Arcane called Precocious Apprentice which gives you one 2nd-level Arcane spell (and only one) which you can take as you take your 1st-level in an Arcane spell-casting class (such as Wizard).
The qualification for Mystic Theurge reads:
To qualify to become a mystic theurge, a character must fulfill all the following criteria.
Skills: Knowledge (arcana) 6 ranks, Knowledge (religion) 6 ranks.
Spells: Able to cast 2nd-level divine spells and 2nd-level arcane spells.
It never specifies that you should be able to cast multiple different 2nd-level arcane spells, and therefore it can be argued that being able to cast the same 2nd-level arcane spells multiple times is sufficient to qualify.
With this interpretation, a Wizard 1/Cleric 3 with the Precocious Apprentice feat qualifies for the spells part, and it just becomes a matter of putting the skill points correctly (max skill ranks at level 4 is 7).
Note: in order to avoid buying skill ranks cross-class, either the Divine spell-caster should have Knowledge (arcana) in class, or the Arcane spell-caster should be taken at 3rd level: Divine 2/Arcane 1/Divine +1.
Note: this is the earliest entry I know of, as I do not know any way to apply the same cheesy interpretation to a Divine spell-caster; in any case, skill ranks would prevent entering before level 4, so only 1 level could be gained.