I'm wondering – how would the Romans have pronounced ph if these letters were together in a word like "triumphantes"? Would it be pronounced like the letter f or the letter p? Did the Romans even ever put ph together in a word?
The diagraph ph in Classical Latin represents an "aspirated voiceless plosive," which means that it sounds like the p in the English word pot – a strong p sound, with aspiration (if you put your hand in front of your face, you'll feel air when you say it).
Ph and its relatives th and ch were initially used primarily to transcribe Greek names and loan-words, like Philippus and philtrum, and later entered native Latin words like triumphus.
More details are available in Vox Latina, by W. Sydney Allen, and in this related answer. Allen concludes his treatment of these letters by reminding us that the ph certainly did not sound like the ph in the English photo:
There is no justification for pronouncing the aspirates as fricatives—i.e. as in photo, thick, loch; this is admittedly the value of φ, θ, χ in Late Greek, but it had not yet developed by classical Latin times.
However, toward the end of the period of Classical Latin, instances of ph as a fricative (sounding like English f) do begin to appear, as described here: When did 'ph' start to be pronounced like 'f'?