What counting system did Pythagoras use? Was it base 10? Something else?
Since we don't have any texts by him, we can't say based on any strong evidence. However, as for his contemporaries and their mathematical means, well, the Attic numerals represent the first attested means of Greek numeric transcription and were contemporary to Pythagoras, with the more familiar Greek numerals coming later. The (latter) Greek numerals continue to be used to this day, in the same places one encounters numerical archaisms in any culture (clock faces, ceremonious titles).
The Attic numerals are biquinary, like the Roman Numerals (so '6' is 'Πl", not 'llllll', and not its own character) but are exclusively additive, unlike the Roman Numerals (so '9' is 'Πllll', not 'lΠ').
As Yst points out, we don't have firm evidence on what (if any) numeral system was in use in Samos or southern Italy in Pythagoras' time. However, number terms in the Greek language are perfectly well attested in contemporary texts and going back to the Mycenaean period. They are very firmly and consistently in base 10, as is the case in all Indo-European languages (or nearly all? dammit, there's going to be an exception, isn't there).
For reference, here are the first several number terms with some comparisons in other IE languages.
| English | Greek | Latin | Sanskrit | Lithuanian |
|---|---|---|---|---|
| one | heis, mia, hen | unus | eka | vienas, sa |
| two | duō | duo | dvi | du |
| three | treis, tria | tres, tria | tri | trys |
| four | tessares, pettara | quattuor | catúr | keturi |
| five | pente | quinque | pañca | penki |
| six | hex | sex | ṣáṣ | šeši |
| seven | hepta | septem | saptá | septyni |
| eight | oktō | octo | aṣṭá | aštuoni |
| nine | ennea | novem | náva | devyni |
| ten | deka | decem | daśa | dešimt |
| twenty | (w)(e)ikosi | viginti | viṃśati | dvidešimt |
| thirty | triakonta | triginta | triṃśati | trisdešimt |
| hundred | hekaton | centum | śatam | šimtas |
It'd be hard to avoid being base 10 with that kind of linguistic backdrop. A few standard linguistic developments in Greek should be obvious from looking at these: the transformation of initial k > p, t (and f in English), and the loss of initial s.