Can I just throw oats into the mash tun (all-grain) or my steeping bag (extract) with the rest of my grains?

Yes, so long as they are "quick" oats as opposed to old-fashioned. The difference is that quick oats are pregelatinized to make the starches available for conversion. Old fashioned oats are not pregelatinized and require a cereal mash first to gelatinize them.

Midwest suggests using Simpson's Golden Naked Oats (#U17019). They add a nutty character to the beer and add a very nice, silky smooth, creamy mouthfeel.