Nooo, don't give up pour! 
reliable and tasty carrots are Amsterdam. They also grow pretty straight, rather than tapered, so you get more in your space and easier to get out. You can even do them in tubs! I wait until after frosts, so late May really.
Courgettes are usually pretty easy so long as they are planted out after frosts (we lost a few in 2011 due to being too keen to get them in the ground). Can you get some compost and enrich your soil before popping them in?
We used some organic potato fertiliser on our tatties last year and they were fab.
Leeks are a great staple. We use two types from Sutton seeds - one that you can eat as a baby leek or a big one, therefore you can eat your thinnings and waste none, and one that you can over winter. We still have some in now that were planted in late summer. It helps to motivate us for winter allotment work to be able to pick some winter veg!
But squashes - pah! Although we cook with them a lot, we cannot get them to thrive at all. I'm not doing them this year as we want to use the space for a nice sweetcorn crop, but i'll be asking the old boys on the site how they get theirs going.
Good luck!