The three grey stripes on the bottom of the cooler is the thermal paste. Nothing else is needed.
Play with the push pins before installation, so you learn how they work.
You must install the cpu cooler with the motherboard out of the case.
This is so you can look at the back to verify that all 4 pins are completely through and locked.
Lay the motherboard flat, perhaps on the foam pad that may have come with the motherboard.
It may take a bit of pressure to push the pins in, and you don't want to bend the motherboard.
The key to getting it right is to push down on two diagonal pins at the same time.