Dark Gifts
Since they no longer have Gnosis, Abominations no longer have actual Gifts, nor can they learn them from spirits or living Shifters. Instead, Abominations have powers known as Dark Gifts. Dark Gifts are corruptions of the Gifts they had in life, warped by their undead natures and inherent connection to the Underworld. Dark Gifts are turned toward affecting the dead, the Dark Umbra, and producing entropic effects.
One examples of a Dark Gifts is Death's Caress, a corruption of Mother's Touch; it can heal the undead and wraiths, but now causes harm to living creatures. Parting the Velvet Curtain becomes Parting the Shroud, which allows the monster to bring others with her to the Shadowlands. Create Element creates plasmic objects and barrow flame.
The mechanics of some Gifts may remain unchanged: Sense Wyrm, Truth of Gaia, Might of Thor, Speed of Thought and those Gifts that cause direct damage (Coup de Grace, Fatal Flaw) and affect the mind (Glib Tongue, Persuasion). However, the thematic elements of these powers change to reflect the Abomination's nature.
Dark Gifts have a Thanatos cost equal to the corresponding Gift's Gnosis cost, cost any Willpower or Rage they normally cost in addition to Thanatos, and follow normal rules for that Gift (required rolls, etc.).
Learning New Dark Gifts: With your discretion, Storytellers, Abominations can learn Dark Gifts from other Abominations or possibly wraiths -- never from any sort of spirit or a Changing Breed. They learn all Dark Gifts at a cost of level x5, regardless of their rank before Embrace. No Dark Gift above level three may be learned after Embrace. Some wraiths may teach Dark Gifts, relating to the Arcanoi they possess; for example, a wraith with high Argos might teach an Abomination Speed of Thought or Leap of the Kangaroo. However, such opportunities to learn new Dark Gifts are very rare and probably won't happen (so if someone is playing an Abomination for the kewl powerz angle she's better off concentrating on Disciplines).

Disciplines
Abominations learn and use vampire Disciplines, but not as easily as other vampires. They don't have "in-clan" Disciplines (which are raised at current x 5), but instead have affinity Disciplines which can be learned and increased without instruction from true Kindred. All Abominations have affinities for the following Disciplines: the three clan Disciplines of the Abomination's sire excluding any sort of blood magic, Fortitude, Potence and Protean. All Disciplines are current x7 to raise, both affinity Disciplines and others. So a Giovanni Abomination (like that would ever happen!) would have the following affinity Disciplines: Dominate, Fortitude, Potence and Protean; not the absence of Necromancy, as it's blood magic.
Abominations don't start out with an initial Discipline point spread, but must purchase the first dot of affinity Disciplines at 10 exp apiece. Abominations are considered the very lowest form of Caitiff, and it's hard for them to find Kindred willing to teach them new Disciplines. (Unless they take care to pass themselves off as "normal" vampires.) However, the potential for learning and increasing their affinity Disciplines is already there; all they need do is spend the requisite experience.
Some Disciplines work differently for Abominations due to their bizarre natures. Unless stated otherwise below, one can assume a Discipline works the same for Abominations as for vampires (though the Golden Rule trumps this).
Animalism: When used on an Abomination, Quell the Beast (●●●) reduces the subject's temporary Rage by one per success on the Manipulation + [Empathy/ Intimidation] roll. (Storytellers might want to apply this effect to living Changers as well.)
Auspex: When used to read the aura of an Abomination, Aura Perception (●●) will register the subject as something out of the ordinary, with vibrant but diseased-looking colors swirling through the pale shades normal for vampires. The reader won't immediately deduce what the Abomination is, unless he happened to previously read the aura of a known hybrid.
Celerity: Yes, Abominations can use Rage and Celerity in the same turn. Doing so, however, will automatically send the monster into a raging frenzy unless a Willpower point is spent; in this case, the character is allowed a Rage roll to see if frenzy occurs.
Fortitude: An Abomination can use Fortitude to soak aggravated damage delivered by silver, fire, sunlight and Wyrm toxins; this is the only soak allowed against these sources.
Protean: Protean comes easily to Abominations, even those not sired by Gangrel. Those who attain the Shadow of the Beast (●●●●) can shapeshift to any of their natural forms with a standard Stamina + Primal Urge roll and no blood point or Rage expenditures; however, they can't assume any additional shapes (like a bat, or a wolf for an undead Mokole), as their forms are already set.
Vicissitude: Most Abominations avoid this unclean Discipline, as they reek of corruption as it is and have enough going against them. As if it needed to be said, statistic bonuses from Horrid Form (●●●●) DO NOT stack with those for the Abomination's natural forms. However, if Horrid Form is taken while the monster is in a form other than Homid, the highest bonus for Physical Attributes can be taken, though all Social Attributes drop immediately to zero. For example, if a Garou in Crinos form (Strength +4, Dexterity +1, Stamina +3) takes Horrid Form (+3 to Physical Attributes), his final adjustments would be Strength +4, Dexterity +3, Stamina +3. Still pretty damned scary.
Blood Magic: No Abominations possess an inherent aptitude for blood magic, since sorcery is based on formal instruction and rigorous study. And it's a safe assumption that no sane or self-respecting blood sorcerer will share his clan's most coveted secrets with an outsider, even if he doesn't know said individual is an Abomination, unless compensated very well. But purely in the realm of theory...
...an Abomination thaumaturgist would probably seek out the Path of Spirit Manipulation (possibly as a primary path), and pick up its principles quickly. Necromancy would grant an Abomination an enormous edge in her dealings with wraiths and the Dark Umbra, though this would effectively paint a bull's eye on her back for every ghost out there. ("Wait... this freak eats us, and can control us and find our fetters? Fuck that!") Koldinism would also fit an Abomination thematically, as it deals with the spirit world... though traffic with the sort of Banes that Tzimisce call up would pile on the Wyrm taint like no one's business.

Oh, you were a Bubasti wizard or one of those other Shifters lucky enough to be Embraced with sorcerous knowledge? Sucks to be you, because now you're a vampire. Your Shadow Craft has fled you; you're treated as a Kindred for purposes of sorcery, which means you only gain the benefits of your previous powers if you're formally instructed in some form of blood magic (as explained here). (Covering my bases pretty well here, aren't I?)