These are the styles of which Kakara is presently aware, and consequently is capable of learning.
Goku Style: The iconic style of your House, this style relies on overwhelming physical assaults and is heavily offensive in nature, falling back on mobility to evade attacks only when severely outmatched. Ki blasts are used, but generally not as a primary feature of combat. They're finishing attacks, not the main show, and outside a special few, practitioners of this form will have little facility here. Given its progenitor, it is not remotely suited to fighting with anybody else, having been created in expectation and even anticipation of fighting all enemies alone. Involved techniques: Hand-to-Hand [Dueling], Kaio-Ken, Kamehameha, Instant Transmission, Spirit Bomb (Lost Technique). Penalized techniques: Hand-to-Hand [Team Fighting], Ki Projection [All Save explicitly Involved Techniques].
Vegeta Style: The iconic style of House Vegeta, this form tends towards a different approach. Created by a warrior who commonly fought entire armies unaided or with limited backup, this form shines when put against multiple foes. As is only sensible for those kinds of conditions, ranged combat proved to be most efficient, and thus while the practitioners of this style can certainly fight up close, they have a lot of talent at range. It is overwhelmingly aggressive and unrestrained, and has virtually no response to being put on the back foot. This style emphasizes absorbing punishment over avoiding it, and thus is not for the weak of body or spirit. Involved techniques: Hand-to-Hand [Crowd Fighting], Galick Gun, Big Bang Attack, Final Flash, Ki Projection [Basic Attacks]. Penalized techniques: Flight, Ki Control, Hand-to-Hand [Team Fighting].
Krillin Style: A balanced form that plays well with others, this style is near-unique in that it seems to have been built in the expectation of being outmatched. Emphasizing evasion and targeting of vulnerabilities, this form has virtually no defense to direct confrontation. Given its creator, that's sensible. Practitioners of the above two forms tend to study this at least a bit so that they're not so helpless should they be confronted with a superior opponent, and so they don't commit the grand idiocy of being so overwhelmingly offensive that they prefer to allow themselves to be punched in the face if it allows them to do the same to their opponent. Involved techniques: Ki Control, Solar Flare, Kienzan, Flight, Hand-to-Hand [Team Fighting], Kamehameha. Penalized techniques: Hand-to-Hand [Dueling], Hand-to-Hand [Crowd Fighting].
Piccolo Style/Demon Style: The form of Piccolo, this emphasizes precise and clever strikes with great concentration of power, and a balance of offense and defense. It possesses powerful techniques at any range, but lacks specialization. Practicing this style demands an ability to control the field of battle as is most expedient at any given moment. Involved techniques: Makankosappo, Explosive Wave, Evil Explosion, Masenko, Light Grenade, Hellzone Grenade. Penalized techniques: None.
Turtle Style: Roshi's style is largely an anachronism that was designed before the full potential of ki was realized by the Z Fighters, and you should not use it in a real fight. It is meant to be used by a fighter locked to the ground, against a fighter locked to the ground, and treats any and all external use of ki as a flagrant and extravagant expense to be saved only for the direst of eventualities. That said, as a foundation for other styles, it's fantastic. Its reliance on hand-to-hand means that practitioners are very good at it in absolutely all of its forms. Involved techniques: Hand-to-Hand [all, signature], Kamehameha. Penalized techniques: All Ki Talents (save Ki Control) and Ki Projection skills, and cripplingly so.
Trunks Style: While Trunks of our timeline contributed little to fighting given that he never actually refined his technique beyond what he had at eight years of age, his future counterpart contributed a brutally pragmatic and survivalist style to the Z Fighters' understanding of combat. This style assumes constant unfavorable numbers against overwhelmingly powerful opponents, demanding insane agility and swift exploitation of exposed openings, with none of the acknowledged theatricality of other styles. In addition, it is the only form developed in conjunction with a weapon. It shines against multiple opponents and when outmatched, and its use of a sword tends to take opponents off-guard. Involved techniques: Sword Use, Hand-to-Hand [Crowd Fighting], Flight (signature), Burning Attack, Masenko, Heat Dome Attack, Final Flash. Penalized techniques: Hand-to-Hand [Team Fighting, or all when without a sword], Ki Sense.
Yamcha Style: This is nearly extinct, but to be honest, it's undervalued. While he eventually retired and was constantly outmatched in strength, many forget that Yamcha was a talented martial artist, who even after years of retirement and with a total lack of Saiyan biology was able to defend against Golden Freeza's invasion of Earth right along with the rest of the Z Fighters. As a derivation of the Turtle School it contains all that you would expect, along with a focus on flurry strikes and esoteric ki techniques that overwhelm the opponent's ability to keep up -- very much the school of death by a thousand cuts. Involved techniques: Hand-to-Hand [Dueling], Spirit Ball, Kamehameha, Flight, Wolf Fang Fist. Penalized Techniques: Ki Sense, Ki Projection (all save involved techniques).
Tenshinhan Style: This form is rarely-practiced, but deceptive for all that. Tenshinhan's form is refined and balanced, offering few concrete advantages, but unlocking the potential to pour the user's life force into an attack. Practitioners have no stylistic weaknesses, and gain a risky but powerful last resort option, in addition to a variety of strange techniques. Involved techniques: Solar Flare, Multi-Form, Four Witches Technique, Kikoho, Dodonpa, Shin Kikoho, Flight.
Gohan Style: A blend of the Demon and Goku Styles, this form is more heavily geared towards hand-to-hand than the Piccolo Style and loses on some of its versatility and most of its precision, but lacks the near-total weakness of the Goku Style outside of zero range. In addition, it is less useless on the defensive than the Goku Style in exchange for tending to be less effective on the attack, and is in fact capable of, when necessary, dodging. That said, it retains a preference for the offensive in a fight. This is an excellent form for fighters with a hand-to-hand focus who don't wish to risk the notorious overspecialization of the Goku Style. Involved techniques: Kamehameha, Masenko, Hand-to-Hand [Dueling]. Penalized techniques: Hand-to-Hand [Team Fighting], all Ki Projection talents that involve the dense concentration of power.
Satan Style: NO. Involved techniques: Losing. Penalized techniques: Self-awareness.
Freeza Style: Overwhelmingly offensive and lethal combat style. This style favors single-target techniques, grappling, and hand-to-hand combat, but has provisions for larger blasts. Freeza stylists suffer (arguably) from one unique flaw -- unlike other martial artists, Freeza stylists must continue to make checks against accidental lethal damage when using a combat skill that has been improved to "Competent" or above if trying to do so non-lethally. Freeza stylists are outright disallowed from attempting to inflict non-lethal damage with skills that are at "Unlearned" or worse. However, using techniques with lethal intent nets a moderate bonus to all checks. Freeza stylists who reveal their study of the style instantly gain the reputation, "Legacy of Freeza," lending crushing penalties to interactions within Exile Society and disabling all positive reputations while among Garenhuld saiyans for a year following the revelation of the stylist's focus. At the end of the year, depending on the stylist's actions, they may or may not get their positive reputations back. Involved techniques: Death Ball, Death Saucer, Death Wave, Death Cannon, Death Beam (signature), Hand-to-Hand (Dueling). Penalized techniques: Ki Control (extreme), Ki Sense, Hand-to-Hand (Team Fighting)