Excellent question, and not an easy one to find a definitive answer for.
First, let's assume that by "first printing press" you mean the first movable metal type printing press in the western world, i.e., Johannes Gutenberg's press of circa 1450.
Asian countries, notably China, had printed wood blocks and even movable wood type centuries before Gutenberg did so in Germany. This would be the speed of the first printing press, but the speed is still unknown.
Gutenberg's press was most likely a converted wine press, capable of printing a single-sided sheet approximately 18" x 12" -- a single "page" would most likely be half of that sheet (his 42-line Bible has a 2-page spread of 445mm x 307mm).
The squeezy bit of the press (the part that applies the pressure to the paper and type) was about half the size of the total paper area. The type needed to be inked by hand, and the paper or vellum being printed needed to be inserted into a holder before going into the press. If we're talking about printing a book using the full sheet size, that would be two pages of type, set side-by-side, printing one side of one sheet of paper, requiring two press squeezes (half of the sheet is squeezed, then the paper and ink holder is moved, and then the other half is squeezed). Then remove the paper holder, then the paper. Repeat. Not a quick process to write, let alone to do!
A conservative estimate: 200 copies of the 42-line Bible were printed in 5 years (not bound, just printed). Each copy is 1,272 pages. That's just over a quarter of a million pages total, or 50,880 pages per year (in 5 years). Assume a work week of 6 days (which means 163 pages per day), say 12 hours per day, call it 13-1/2 pages per hour.
An aggressive estimate: 2-1/2 years (let's give up half of the time to misprints, mistakes, etc) means 101,760 pages per year. Assume Gutenberg was so enlightened that he only worked 40 hours a week, 50 weeks a year. That's roughly 50 pages per hour.
Call it somewhere in between: 20-30 pages an hour.
For comparison: four centuries (plus) later in 1880, Lord Stanhope had a cast-iron press which produced 480 pages per hour.
Within a 50 years of that, platen presses were producing 2,500 pages an hour.
These days, there are offset web presses capable of 100,000 pages an hour -- and those "pages" are about eight times the size of Gutenberg's, printed both sides, so by comparison: today we can print about 114,000 times faster than ol' Joe.