The key to quantum field theory is the Feynman diagram, which comes from the Feynman path integral formulation of quantum mechanics (when applied to classical field theories). Thankfully, though, the mathematical rules associated with Feynman diagrams can be derived from slightly simpler methods than using the path integral (or I sure wouldn't be telling you about finding them). I'll put in a little about how to do that later, when I get the chance. The mathematical formulae associated with the diagrams add up to give you the Feynman amplitude, and you can use that to calculate cross-sections for collision events, decay rates, etc. I'll also put in how to do that later.
In the field theories that we know how to work with, vertices in Feynman diagrams correspond to multiplication by a small constant, so we can safely ignore Feynman diagrams with more than a certain number of vertices (to a certain order in perturbation theory). Typically (unless you have a theory with a number of different kinds of vertices), the lowest order in perturbation theory consists of tree diagrams, and the higher orders are diagrams with loops, so the orders in perturbation theory correspond to tree order, one-loop order, etc. Then all you have to do is evaluate the terms out to a certain order, and you're done. However, Feynman rules for loops include integrals over all of 4-momentum space, and those integrals usually don't converge. What do we do? We go to the next section.