There are thousands of free VST effect modules you can download and experiment with, and billions of combinations of using them. Some of the classic units you should consider:
Compressor (Not VCA, optical compressors are much brighter and more musical)
Tremolo (Found on classic Electric Pianos)
Amplifier (Tube amps gave volume, but also colour)
Overdrive (Valve amps or Stomp Boxes give a great 'dirty' effect)
WahWah (AKA Envelope Filter, wasn't just guitarists using these pedals)
Phaser (Sweeping, bubbling effects, common on Electric Pianos)
Chorus & Flanger (Enriches and fills sounds)
Delay / Tape Delay (Classic 'Tape Echo')
Spring Reverb (Hammonds had built-in springs, and they were widely used elsewhere)
Reverb (Simulates the effect of playing in rooms, handy for 'expanding' sounds)
MULTIPLE EFFECT UNITS
Effectizer (Soundfont Legacy Pack)
Website
Effects: WahWah, Overdrive, Tremolo, Modulation, Stereo Delay, Reverb
Great collection of effects, designed for organ and electric piano instruments. All six units can be used at once, in one set order.
LFX-1310
Website
Effects: Filter, Compressor, Amp/Overdrive, Modulation (many), Delay (many), Reverb (many)
Fantastic multi-effects unit, three units can be used at once in any order.
EFFECT UNITS & BUNDLES
NuBi Plus Pack
Download
Effects: Crossover Delay, Phaser
Two great units, originally designed for NuBi organ.
Blockfish (Fish Fillets Pack)
Website
Effects: Compressor
Really high-quality optical-like compressor with "soul", unlike most guitar compressors.
Endorphin
Website
Effects: Compressor
Vintage-style opto (photo resistor) mode, with classical feed-back detector circuit.
PianoVerb
Website
Effeccts: Piano-specific Reverb
A professional reverb unit, designed for acoustic pianos but sounds gorgeous with EPs.
Retro/Tape Delay
Website
Effects: Delay, Tape Delay
Stunning, in-depth, delay effects.
Variety of Sound
Website
Effects: Vintage EQ, compressor, tape, colours, reverb.
Large range of vintage effect units. Gorgeous sounds, gorgeous interfaces, easy to use.
TAL
Website
Effects: Filter, Reverb, Flanger, Phaser, Chorus (many more)
Very high quality, lovely vintage visuals too.
Classic Effects
Effects: Filter, Chorus, Compressor, Delay, Flanger, Phaser, Reverb (many more)
Very large range of effect units.
MDA
Effects: Amp, Optical Compressor Delay, Detune, Filter, Overdrive, Flanger (many more)
Large range of effects, text-based interface.
Effects: Autopan, Compressor, Phaser, Modulators, Tremolo (many more)
Very high quality professional effects, but "nagware", sometimes opening their website.