Yes, but prioritize running a fast mile before the 5k.
At any run group I've been to, I can always tell who ran in high school/college vs. runners who started running seriously as adults. The reason is that most people who run in (USA) high school will train primarily for a middle distance event (800m-3200m) early on, even if they end up doing marathons and trail shuffles as an adult, whereas starting as a recreational runner later in life, you focus on longer distances right away. Runners who have never trained for a mid distance event seem to have less power in their stride, and also seem to prefer a slower pace for everyday running.
Every new runner who cares about PRing should spend 6 months training for the Mile* starting within the first year.
* Or 1500, depending where you're from