As you already know, A good spring reverb (or emulation) is an essential part of the sound. The reverb accentuates note attacks a little like a slapback delay in a way that's often described as 'drippy'.
Use of a guitar with a whammy bar is also important. Chordal and melodic parts alike are often quite elaborately and dramatically inflected with the whammy bar, allowing dips in frequency (in contrast to string bending, which pulls your frequency higher).
Tremolo picking is a trademark of some surf musicians.
The three aspects above - 'grungy' reverb sound, exaggerated use of whammy, and aggressive, sometimes tremolo playing - contribute to the ability to get a sound which is 'dirty' without being distorted. In fact, a clean amp sound is quite important to allow the note attacks to come through.
Surf music was one of the first electric-guitar based genres to enthusiastically explore territory away from the major scale. Not all surf music is minor but minor chords and scales are common, as is venturing into more exotic territory like the the Double Harmonic Major Scale in the traditional song "Misirlou" given its famous rendering by Dick Dale.
Simple riffs and repeating figures to animate baselines and accompaniment parts are also common.
One possible issue if playing solo is that surf music is probably more suited to full-band arrangements than solo guitar. Its focus on simple lines each played with energy and expression makes it quite easy and fun to play one surf part on guitar, but unlike folk or country, it's hard to make a full arrangement for a single guitar that still has all the trademarks of 'surf'.