Looking at schedules on the Trenitalia website, it takes less time to go to Naples (Napoli Centrale), and there are more direct trains. There's a 6:07 direct that can have you there in 1 hour, 45 minutes, and a 7:27 that does it direct in 1 hr, 59 minutes.
All trains to Salerno involve at least one change and take a minimum on 2.5 hours or more.
Tickets are inexpensive for either option.
Explore your options on the Trenitalia website: http://www.trenitalia.com/tcom-en
Along with taxis, you can also get to Positano from Salerno by boat if it's not a rough-sea day when they can't dock.
Here's the current ferry schedule:
http://www.positano.com/en/ferry-schedule?from=salerno&to=positano
Take a cab or walk to the port. There are no direct ferries from Naples to Positano although it can be done via an indirect route, like Napoli to Sorrento or Capri and then to Positano.
There is also this shuttle bus service from Naples although based on some of the past reviews, I personally wouldn't trust it.
http://www.positanoshuttle.com
http://www.positanoshuttle.com/positanoshuttle-bus-timetable-2/
You could also take the Circumvesuviana commuter train from Napoli Centrale to Sorrento and a driver from there.
If you are a group (I think you might be, based on past posts?) or aren't concerned about economizing, then a driver from Naples might be your best option.