[Potentially incomplete answer. I have only watched the anime and read "Tsubasa Song", and I suspect that there would be more information in the light novels. Some referencing from the Wikia entry for Araragi, since I needed something to jog my memory.]
Through-out the series, we see a couple different traits that were presumably acquired as a result of having become a vampire:
Firstly, Araragi has various healing powers. For instance, even after being attacked by staplers in the beginning of Bakemonogatari, it seems that his mouth was able to heal very quickly. Moreover, in Nekomonogatari Black, he heals Hanekawa's wound with some of his blood.
In the Hachikuji arc, Araragi reads Hachikuji's name off of her bag. I don't remember the scene very well, but Senjougahara is unable to see the name. Araragi is not aware -- at that point in time -- that Senjougahara is unable to see Hachikuji. Rather (per a discussion on TV Tropes that I read -- see "one dialogue, two conversations" -- and my memory of the scene), he assumes that he was able to read the name because of some enhanced vampiric vision.
In Nisemonogatari, we see Araragi being hesitant to spar with his sisters. This is because he doesn't want to harm them because of the super-human strength he has acquired.
Later, Araragi removes some of the poison that has been making Karen feverish. However, it's not clear if this is just a vampire ability or a result of the fact that given that the fever is supernaturally generated, he can remove some of it. Without more information, I would assume that it's probably the latter.
Most definitely at least during the events of Bakemonogatari in the in-universe chronology, if not after it, Araragi has a conversation with Meme Oshino (or possibly with Shinobu -- I don't remember which).
He is told that he has the option of becoming entirely human again, provided that he does not feed Shinobu his blood. Moreover, in "Tsukimonogatari", we see that Araragi is slowly becoming a vampire -- he can't see his reflection anymore, and Shinobu closes the blinds in his room to prevent him from contacting sunlight.
The above suggests that the exact abilities Araragi has are dependent on how much blood Shinobu has been allowed to suck from him -- we see, for instance, in "Suruga Monkey", that Araragi preemptively lets Shinobu suck blood from him to ensure that he won't straight-out die from being attacked by Kanbaru. Furthermore, Araragi's exact abilities eventually change.
But in any case, normally speaking (i.e. when he hasn't let Shinobu suck a lot of his blood), Araragi's abilities seem to essentially be slight improvements of normal human physical abilities (of sight, strength, healing), and nothing more than that. For instance, unlike Shinobu:
Araragi is incapable of time-travelling alone -- it's Shinobu who creates a portal to the past in "Mayoi Jiangshi" -- and we never see him travel at really high speeds when he is "normal" (whereas Shinobu presumably did when she escaped with a man to Antarctica).
