Right now my playing is limited to playing pairs of strings that are right next to each other. What's the best way to expand to larger intervals or three string patterns?

The first thing I teach my students is to play a wide range of different grips. Grips are the combination or strings you "grab" and pick at the same time. Try different three string grips.

Three adjacent strings together.
Two fingers on adjacent strings together and the thumb on a string one string down from finger.
Three finger grip with a string skipped in between each pick.

There are two main reasons to learn these different grips. One is to learn different "voicings" for the same chord so that you can play a C chord as CAG or AGC or GCA. The second reason is to leave out notes that don't belong in the chord.

Understanding the notes you're playing and what purpose they have and why you would play them will designate what grips you would use.

-- Ricky Davis