A spade - be careful that you choose a handle that you feel happy with as some people like the Y shaped handles and some people work best with a T handle
A fork - again choose a handle that you feel happy with and you want a wide blade for general digging but may want a tattie fork if you are going to grow a lot of spuds (thinner blades)
Now I'm crazy and can only work with a T handled spade and a Y handled fork for my own comfort - silly thing but do get a tool you are happy with.
A hoe - there are a whole variety of hoes and you have to work out which one will suit you; I always use a
swoe and wouldn't know how to use any other sort of hoe after 25 years
And rake - you wouldn't believe how many rakes there are either!
Have a look here and decide what you are going to do with the rakeDefinitely a trowel and small matching fork; also a small pruning knife will do many small jobs and a penknife is endlessly useful for cutting twine and opening bags, etc, a couple of pairs of small light snips for cutting peas and beans off, collecting herbs and soft fruit
Hammer and nails will definitely come in useful; don't forget to get yourself some garden twine; also you may need that plastic covered garden wire to build supports for fruit canes; long handled loppers can be darned useful too if you have hedges or bushes.
Oh and I want
one of these for marking out rows! All the family gardening uncles had one when I was a kid - so useful.
I worked for 3 months on the allotment and then went and got a cheap wheelbarrow so that I could shift compost around more easily.
And don't forget the shovel - you just wait till you have to move compost and then you will know why you need a shovel.
Water butts are essential and so is a good shed for storing the tools.
Freecycle is an excellent place for tubs and pots - you can grow so many things in bigger tubs tucked in a corner.
Hosepipe and fittings - if you can get one on a reel even better, watering can, heavy duty black plastic for covering things like compost heaps, planks of wood for building compost bins and raised beds, netting to cover your smaller crops against the thieving birds, lots of canes of various sizes for holding netting in place as well as for supporting crops like peas and runner beans.
Old bricks for holding things down come in very handy. I have hedges and the best thing that I've been given was a pair of free, second hand shears with wooden handles - an old style but when sharpened they have become a very good friend.
Sure it's a lot of tools and it's taken me two years to collect them but every one is essential - I do have a double allotment mind.