Clothing: Stout boots or Wellington's for digging, pair of gloves to protect your hands when needed and a wide brimmed hat in sunny weather  or a woolly hat when its cold.
Equipment:
Spade: used for digging and turning soil, will also be required to remove turf  prior to digging.
Fork: for digging over soil and digging in manure and the like. Will be required to dig out root crops and weeds. Hold the tool vertical, tread to the top of the blade with the ball of the foot, lever the shaft backwards and lift and turn the soil.  
Rake: for levelling soil and preparing seed beds.
Dutch hoe: used to cut down  annual weeds  and prevent compaction of he soil surface and water evaporation. The method is to hoe the area in front of you, moving the tool towards you, step back a pace or two, then hoe the next  area in front of you. Continue in this manner.
Draw hoe: Used primarily to earth up plants, dig seed drills and for weeding. To weed or cultivate soil the method is similar to the Dutch hoe, except the cutting action is move the tool  forward over the soil to be worked. You do not walk over the hoed soil, but step back a pace and continue as before.
Secateurs : For pruning bushes, trees and cutting flowers.
Knife.: A sharp one for cutting twine or plants.
Hand trowel.: needed when transplanting.
Hand fork: used to dig out or loosen weeds prior to hand pulling.
String line: Needed to mark out rows.
Large watering can and a bucket. The bucket can be used to refill the can or collect up weeds. A watering rose for the can is advisable for watering seedlings.
Pressure sprayer: For the application of insecticide and fungicide.

Suitable second hand tools can be gotten from boot fairs. Expensive shiny tools are not advisable, shed robbers look for these, in fact leave nothing of consequence or value in the allotment shed, take it home with you.