A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A fan is a finite set $F$ of polyhedral cones, all living in the same vector space, such that
(1) if $\sigma$ is a cone in $F$, and $\tau$ is a face of $\sigma$, then $\tau$ is in $F$.
(2) if $\sigma$ and $\sigma'$ are in $F$, then $\sigma \cap \sigma'$ is a face of both $\sigma$ and of $\sigma'$.
This blog post of mine might help you visualize these definitions.
Most mathematicians I know learned fans from Fulton's Toric Varieties. This would involve learning a lot of algebraic geometry on top of your combinatorics, although it is algebraic geometry that is very relevant to tropical geometry.
For a pure combinatorics reference, have you tried Chapter 2 of De Loera, Rambau and Santos? They focus on polyhedral complexes, which is the more general setup where you don't require that the half spaces pass through $0$, but they talk about fans as well. I haven't had a chance to look at it yet but, based on my knowledge of the authors, I expect it is very good.