Space Punks, the four-player looting shooter, does things a little differently. Instead of allowing you to allocate Talent Points whenever/wherever you wish, you have to go to a specific vendor.
Talents Points are acquired upon earning experience to gains levels. Once a level has been gained, you will earn Talent Points to spend on the very in-depth Talent Tree Space Punks have to offer. But you cannot spend them straight away.
To spend Talent Points, you have to play a few missions of the game until the Skills & Talents vendors open up the hub area. The Skills & Talent shop can be found to the east of the hub area.
Here, and only here, can you spend your Talent Points. Interact with the vendor, and press RB or E, and from your character's avatar at the bottom, you can select any of the branching trees.
Even upon changing character, because you have unlocked this specific vendor, they are always available. You cannot allocate your points wherever you are or even during a mission. You have to wait until you are back in the hub area to do this.
This is a bit of a strange system, especially when it resembles Borderlands so much, but alas, knowing where to go is half the battle! You can come to the vendor when you have a few Talent Points, or you can save them up. It's up to you!