If it's hot where you live or if your kitchen is too hot, don't let the butter or cream cheese soften at room temperature before making the buttercream. However, if it's cold then leave them outside of the fridge for a couple of hours before starting.
Mix the butter using the mixer attachment in your stand mixer for 5 minutes from the moment it goes soft or until it looks really pale. Remember to stop every once in a while to scrape down the sides and bottom of the bowl if you're using a stand mixer.
Add the cream cheese and vanilla and repeat the process.
Add the sugar using a sieve and mix until everything is incorporated.
Spread the cream cheese buttercream on the cooled-down cake. It's important it's no longer warm because it could melt out buttercream.
Place the cake in the fridge for at least a couple of hours before cutting so the buttercream is easier to cut.