In most cases where a Discipline is used against another character, the person who calls their Discipline is usually the aggressor and thus the victor. See; Dominate vs Willpower, Auspex vs Obfuscate [both Dominate and Auspex win in these situations]
Advancing a Discipline (or learning a new Discipline) is done by mentioning in a Downtime email that you wish your character to start learning the discipline in question (as well as how they learn it, if it is Out-of-clan it will require a teacher, and Caitiff require a teacher regardless) and takes a month for Vampires, or three months for a Ghoul. For vampires, the cost is 3xp per level if the discipline is in-clan, or 5xp per level if it is not within their clan (ie. a Malkavian going from Auspex lv2 to lv3 would be 6xp, but going from Potence lv2 to lv3 would be 10xp). Caitiff have slightly different mechanics, where the costs are a flat 4xp per level regardless of the Discipline. For Player-character Ghouls, they may only learn whatever Disciplines their domitor Vampire has (and only up to either Lv3 or the level their Domitor has, whichever is lower) and pay effective Out-Of-Clan costs for them (after Character Creation, during creation when spending Freebie points they spend as if the disciplines were In-Clan, but are still limited to the other Ghoul limitations listed above).
When learning a new In-clan discipline the cost is a flat 5xp for the first dot, or for Out-of-clan Disciplines the cost is a flat 7xp. Caitiff spend 6xp on the first dot of any Discipline. Ghouls spend 7xp on the first dot of any new Discipline, assuming their Domitor has that discipline.
NOTE; All of these rules are to be used to determine how your character should act when using them, or to understand how your character should react to a power being called. The actual mechanics of how they work is not considered to be IC knowledge – instead, if you wish to use information about how the powers work from an IC standpoint (perhaps to explain to another character IC how the power works, or to plan out your own use of the power in Downtimes etc.) you should refer to the Lore skills your character has as well as if they have the power in question themselves.
Additionally, sometimes there will be certain situations where a power’s effect works differently, as dictated by a Narrator or NPC at the time. This is specifically for plot reasons, and as such may not act the way these rules state the power does. If you’re unsure whether a call was made for plot reasons or if an NPC/Narrator simply got the rules wrong, drop OOC (by putting your hands in the T-symbol) and ask a nearby Narrator for a clarification.