Oh right, no problem.
Touko Travel is... well it's more or less just a method of travel developed by a character by the name of Aozaki Touko, somebody who is more relevant in other Type-Moon works, namely Kara no Kyoukai and Mahou Tsukai no Yoru. If I have to be completely honest with you, the entire character of Mitsuki was basically just an excuse to give the joke that is Touko Travel some time in the spotlight because I grew fond of it somehow, even though I think it was only ever a comment made by somebody in passing xD
Anyway. In all seriousness, there are three methods to travel by broom. One is the Jet Method, which is pretty self-explanatory, then there is the Ether-Sail Method which is basically just a glorified gliding ability, and the last is Touko Travel, which works by placing a Magical Anchor and pulling yourself towards it. That's basically it. There's nothing more to it than that, but I guess this is just another example of a trivial idea entertaining me to the extent of creating a character based around it... so yeah... take that as you wish ^^
In any case, then there is the issue of Origins. Phew, where to begin...
You were pretty spot on with your own definition, but there's a little more to it than that which might be harder to grasp if you haven't watched the movie series Kara no Kyoukai that is set in the same universe as Fate. I guess Fate/Stay Night also deals with Origins to an extent, but even then, it is built on the understanding that was introduced in Kara no Kyoukai. Anyway. I'll try my best to summarize the key points.
An Origin is defined by the Mages Association to be a "Compulsory Driving Force from within the 'Root' that has managed to escape and take on Material Form". While a Person's Origin can remain dormant for the majority of his life, once awakened, it is impossible to go against one's Origin. Rather than being forcefully subjected to the will of one's origin, one's own thought process and instincts will naturally incline to follow one's Origin similar to how one would simply act on his Ideals and Beliefs. In terms of Thaumaturgy, while a Magus' Elemental Affinity is useful for determining one's general alignment, the knowledge one's Origin grants one unhindered insight on his opponent's nature, behavior, ideals, and even his abilities to an extent. While rare, there are cases were a Magi's Origin is expressed to an unnatural degree, replacing his Elemental Affinity altogether and limiting him to Thaumaturgical Processes that align with his Origin. While this severely restricts one's ability to invoke conventional magic, Mages with such qualities usually tend to specialize on a singular field of Magecraft to a masterful degree, and often surpass the capabilities of conventional Magi in their selected fields. A perfect example of such a case would be none other than Emiya Shirou, whose Origin of "Sword" has manifested to such a degree that his mental landscape has become an endless expanse of swords which he can summon via a Reality Marble, giving shape to his ultimate ability "Unlimited Blade Works".
Think of an Origin as the core of someone's being. If you were to take a person and reduce them to their bare essentials, to the raw impulse that drives their actions, that is when you'd discern their Origin. Unfortunately it is a very abstract concept, but I hope this helped. I'm pretty bad at explaining things though, so please feel free to point out when I'm just talking nonsense! I let myself go at times and don't realize that I'm just talking gibberish.