So what do you do?

It the vegetation is really tall, its best to cut it down.  Manually, a scythe or slasher is the best option, or you can ask in your allotment site if anyone has a strimmer and  could they cut your plot for you.
So, pile up all this cut vegetation, near your inherited shed, or near the perimeter of your plot- this is the start of your compost heap
Now you have cut vegetation at or near ground level.

You don't have lots of time for digging, because you want to get pleasure and not too much backache out of this plot.  If you have bindweed, couchgrass or creeping thistle, just digging it over or having it rotorvated wont help in the slightest- every cut piece of root makes a new plant, so you will just make the problem 10 times worse.

The roots will need to be dug out, but this is much easier if the plants are weakened first.
A good approach is to cover the ground with something which excludes light.  Old carpet is often suggested, but the rubber-backed modern carpets do not bio-degrade.   We have found an old unwanted tarpaulin very effective (found on the allotment site).  Black plastic is OK, but does not allow moisture through, so you either get slimy mud or solid concrete underneath the cover.  If you can afford it, mulch-matting is the best as this allows the soil to breathe whilst blocking out light tot the weeds.  It only takes a day to cover half a plot, especially if you can get someone to help.
A covering for 6 months will weaken weeds to some degree, and may be OK if you don't have a huge problem.  But if your plot is like mine, the main plants were the perennial creeping weeds listed above, and we have found that a years cover is much more effective.   The remaining weed roots can then be removed when you work the soil over for planting your crops.  Work in some compost (from your kitchen scraps and waste, keep your compost heap on your plot), or if you can obtain it easily, well-rotted manure is a wonderful option (local stables or farmers sometime have this available)

OK, but while you're waiting you will want to grow some crops, so you will probably have to dig over and remove weed roots from at least a small area of your plot- spend an afternoon doing it, and you should end up with an area of a couple of square metres- enough to start being productive.  Aim to bring an area equivalent to a days weed-removal into use once every three months.
But learn to live with your weeds.  As long as they don't swamp out everything, they can be tolerated.  Dont be put off by the weed-free plots of others.

Beneficial crops are those which don't require a lot of effort- if they can withstand a weekly rota of neglect, they will survive and crop for you and look after themselves.
For salad leaves, choose things like rocket, cornsalad (Lambs lettuce), landcress, sorrel, which can be sown or planted once, and either left to seed themselves every year, or are perennial- this also means that you end up with a year-round supply of salad leaves after a spring and an autumn sowing and seeding rotation- autumn (august/september) sowing of seeds gives a winter crop with no effort.
If you want to try green manures to cover the ground, this might be worthwhile, but if the weeds are robust, they could well overrun the sown crop.
Things which grow quickly early in the spring are a good bet- broad beans will get very tall, and weed seedlings wont get a look-in, while peas will scramble up some old chicken wire supported on poles, away from weeds.  Germinate your seed at home in a saucer, in feb/march, filled with damp kitchen paper till the rootling pokes out, then plant them out (start them on a monday night, and they will probably be ready for you to plant at the weekend).  Plant your germinated (Chitted) seeds about 4cm down, in a hole so you dont damage the emerging root.  You could grow them to small plants in potsin the house if you know you're not going to get to your allotment for a couple of weeks- a transplant has a good chance of survival.
Potatoes are very good in ground that you have had covered through the spring weed flush-uncover in May and plant out a chitted maincrop potato (even in June) and although there will still be weed in there, the potato plants are much bigger- you can then get a lot of weed roots out when you harvest your spuds.
Tender crops which cannot go outside till late May/early June are good.  Transplant them from your window-sill or doorway nursery at home as sturdy plants and they should do good against the weeds.  These crops like beans, pumpkins, tomatoes, chillis, are always good value and are quite resistant to neglect once established, and will go on cropping with luck until cut down by the frost in October.  At this time of year, its good to transplant in an evening after work- watering is more effective at this time of day, and you can do a bit of hand weeding and then pick some some crops for use in the kitchen whilst you're there for your supper after a hard evening down the allotment
Cabbages/broccoli again can be started in pots and transplanted out, under a "cloche" of chicken wire when they've grown up.  French/runner beans for the summer do best on ground that you've had covered for a season, and again, they could be started at home and transplanted.

Slugs will hide in piles of weeds you leave lying around- try and keep to one big pile, and watch for slugs out and about on your evening visits- squash them on sight.
Don't try and get the whole plot into cultivation in one go- you just cant.