What are the fighting styles which employ the katana from the beginning of training?
Major categories that I am aware of (expanding slightly on Sardathrion's comment):
Iaijutsu (居合術) and Iaido (居合道) are the art of drawing the sword. Iaijutsu, theoretically, prefers more practical application while iaido is closer to an internal martial art, but you'll find schools under both names that exist along a bit of a continuum.
Kenjutsu (剣術) is actually a bit of a blanket term for traditional Japanese swordsmanship and is literally "the technique of the sword."
Kendo (剣道) can be more sport/sparring oriented than kenjutsu usually is, and has formalized rules, scoring, judging, etc.
These are all essentially umbrella terms for more-or-less what you seem to be looking for: martial arts that teach katana.
Also needing a special mention here is kumdo (검도 (劍道)), which is a Korean derivative of kendo.
My experience has been that aikido dojo vary hugely in how early and how seriously they integrate weapon work, but you can certainly ask if you have an interest in learning aikido along with the sword work. The same is true of a lot of other martial arts: Some of them integrate weapons early, some later, and the only real way to figure it out is to ask at the individual dojo level.
Many ninjutsu and related arts like To Shin Do have weapons classes open to all from the beginning, some even incorporate weapons training into regular classes as well. Look at your friendly neighborhood band of ninjas to see. :)
Kendo the "way of the Sword"
Iaido the "art of Drawing Swords" or " being Consonantly prepared"
Both teach you the ways of using a Katana but don't necessary require the use of the actually blade to learn. They also focus on the mind set that some with practice the blade and learn how to cut with one.