Does an English Premier League team play its first 19 fixtures of a season against 19 different clubs?

Yes, this is how the games are scheduled unless games get cancelled due to bad weather , ... and are scheduled to be played later on (which usually doesn't happen). The team that hosted the first game will play in the opponents' venue in the second game. This is the conventional schedule for National European football leagues.

EDIT:

So in response to those who downvoted or asked for reference, the following was taken from Wikipedia

There are 20 clubs in the Premier League. During the course of a season (from August to May) each club plays the others twice (a double round-robin system), once at their home stadium and once at that of their opponents, for a total of 38 games

So each 'cycle' mentioned by @Yatizme is a half-season where each team plays the other teams once.