In its usual fashion, DF's Algebra treatment of tensor products is insufferably verbose. The problem I am working on asks me to show that $\Bbb{C} \otimes_\Bbb{R} \Bbb{C}$ and $\Bbb{C} \otimes_\Bbb{C} \Bbb{C}$ are $\Bbb{R}$-modules, but I am not even sure where to begin, and poring over the text hasn't helped much.
I think that most of my confusion lies in the fact that this "problem" doesn't seem very much like a problem at all. I mean, isn't $\Bbb{C}$ an $\Bbb{R}$-module and doesn't the tensor product construction guarantee that it is a module over whatever the factors are a module over. If so, then there is nothing to show, right? I'm confused.
As @Quasicoherent mentioned in the comments above, it seems likely that the main part of the problem was the latter half.
I'd like to stress one point though. While it may be clear that both of these modules admit a left $\mathbb{R}$-module structure, I believe it is important that you pin down where it is coming from. It may surprise you. There are actually several ways these particular objects might acquire their module structures, and it is convenient that all the obvious ones coincide in this case.
You may find the following little exercise instructive: Show that $\mathbb{C} \otimes_\mathbb{R} \mathbb{C}$ admits two different left $\mathbb{C}$-module structures.
If you go on to study noncommutative algebraic structures, it can become quite a headache if you get fast and loose with where your actions are coming from.