If I'm getting the Core Book, do I need the Player's Guide?
There is no information in the Player's Guide that is not in the Corebook. Its use is as additional copy of that part of the material.
If you want to run the game (as opposed to being a player), it can make sense to have a copy of the Player's Guide around. You can leave it on the players' side of the table for their use -- they can quickly check how their ability works or what equipment the want, and won't find any GM information in there.
When generating characters with a whole group, you want as many copies of the Player's Guide around as you can get, plus Character Options. It just speeds up things considerably since everybody wants to read at least some of the Focus descriptions.
Note that in The Strange, this applies to choosing a Focus for a new recursion as well, so have these copies at hand whenever you plan sending them someplace new. If that is not how you want to play it, check out the optional rule on p360 of the corebook.