Which button changes to object mode? I can only find buttons for game mode and build mode.
It's one of the tools. Have you ever placed an object/blueprint before? That was object mode.
The be really really specific: spin your mouse wheel until you see a little chair at the bottom of your screen. Now you are in object mode. Where you can do stuff with objects like place them, drag them around, right click on them to open a radial menu, etc..