A reaction for which no reaction intermediates have been detected or need to be postulated in order to describe the chemical reaction on a molecular scale. An elementary reaction is assumed to occur in a single step and to pass through a single transition state.
So you're right that elementary reactions occur in a single step/have a single transition state. I'm not sure what you mean by "obey the rate law", but necessarily, a reaction that has only a single transition state is first order in each reactant. Because there is only a single step, the concentration of every molecule involved in that step effects the rate.
In practice, because the probability of three molecules colliding together in a way that can cause a reaction is very low, all elementary reactions involve one or two molecules. A single molecule reaction is just a dissociation, e.g. $A_2 \rightarrow A + A$, with a first order rate in A and an overall order of one. A two molecule reaction could be something like $A+B\rightarrow AB$, with still a first order rate in each of the reactants for an overall order of two.
In your question, you've identified that SN1 reactions have multiple transition states, so SN1 cannot be an elementary reaction. In fact, the 1 in SN1 means that the overall order of the reaction is one, but since it involves two reactants, the order would be two for a single step reaction. Naturally, an SN2 reaction has an overall order of two and is an elementary reaction, proceeding through a single transition state where a bond between the incoming nucleophile and the carbon centre is formed at the same time as the leaving group's bond is broken.