You sound in a very similar situation I was when I was given my first allotment. I also did't have a garden and had nothing when I started. To start with I would recommend getting:
Fork and spade. I usually dig with a fork as it doesn't chop up bindweed and couch grass roots like a spade. My new plot was matted with it but saying that a spade is needed to dig out brambles, planting fruit trees etc. Make sure they feel comfortable when buying them (I looked rather silly when choosing mine as 'air digging' isn't really fashionable) but you will be using these whilst digging over your entire plot and comfort is required, not a bad back and blisters!
Twine and bamboo/wood sticks: to mark out things such as paths, plot boundaries etc.
Trug or a large plastic bucket to throw all of the weeds and rubble into. I find filling a 75L trug with weeds a manageable weight for a woman to carry up the plot to the compost pile.
Hoe: Once you've dug your plot over this is indispensable to stop the weeds from growing back.
All the rest can wait. You'll be pleased to hear that in 2 years I have managed to fill 2 sheds with bits and pieces that I use.
Good luck with your new plot
