At the most basic, you will need mask, snorkel, fins and a speargun. For the UK a wetsuit is very useful (some would say essential), you can get away with a surfing suit to start with, but a proper spearfishing wetsuit is much, much warmer.
If you have a wetsuit you will need a weightbelt to conteract the extra buoyancy.
I would also recommend you get a float to tow behind you (so boats don't run you over, and to rest on if you get tired) and a stringer to carry your fish. You should also carry a small sharp knife (a kitchen devil in a piece of hosepipe is ideal!)for cutting snagged line and cleaning fish.

Specific recommendations;

Mask; Wide field of view and low volume. I recommend the Spetton Team mask . Fit is very important, we will exchange or refund any mask you don't get on with

Snorkel; Simple, wide bore, no valves, flexible is best, Flexiplush snorkel Fins;

Long blade fins are much more efficient. Spetton fins . Fibreplastic blades may be added later

Wetsuit; 5mm two piece, no zips, attached hood. Apneasub economy suit , or Apneasub Triton made to measure is well worth it if you have the money

Weightbelt; Rubber is best as it doesn't slide around. Marseilles belt

Speargun; 75cm rubber gun is best for a beginner in UK waters. Sportline 75 , AMI II 75, or Velos 75

Float; Spetton Torpedo float

Stringer; Rob Allen stringer