The 64-bit version of Steam on the Steam website is currently not functioning correctly on 64-bit 16.04 LTS systems (login loop). However, like CelticWarrior said, the 32-bit version from the repository "multiverse" is perfectly functional and supported. Make sure you have the repository "multiverse" enabled, and then:
sudo apt-get install steam
sudo apt install steam
Also, don't fret that you have a 32-bit version of Steam. Games that are 64-bit will be installed as such.
I don't recommend gaming on linux at the moment. Unless you're running a nVidia card, AMD and Intel OpenGL support is terrible and, in general, DirectX, under Windows, is a lower-level, faster, and better API. I dearly hope this changes with the introduction of Vulkan.
I would love to run games like CS:GO properly under linux (doesn't render fullscreen, ~20% fps hit, etc.), as well as future games that may utilize Vulkan. However, I don't see this at the moment. Stick to Windows for gaming. On a side note, it also has easier and more efficient support for gaming keyboards.