as others have siad - it all depends on where abouts you are in the country!
Peppers........1st time out, I'd suggest buying plants that have been started off. Germinating pepper seeds can be a pain - you need warmth & light and they need constant attention! A bit of a pain.
So, buy some that have been started off.
Put them into a nice big pot 10-15 Litres - and then you'll not need to repot them.
They need somewhere light and sunny, so keep them indoors while young - ontop of a radiator on a sunny windowsill, in a heated conservatory
once it's a bit warmer you can move them out to a greenhouse.
they like a bit of tomato feed
That should do you. (you can also do chillies this way and a good starter plant for a good harvest is an Apache chilli plant)
Carrots are pretty easy - get a good compost, something that is quite low in nitrogen (you want to avoid something that has manure in it - as the nitrogen content will lead to "that's Life" shaped veg!! Which can be amusing!
wait until the frosts are all passed. They are best grown in a raised bed, as the soil isn't as compacted and it's easier for them to grow down - pushing soil out of the way. If you are not using a raised bed, dig over the ground and make sure it's well tilled
you can do it 2 ways - sow your seeds quite close together - and "thin" them out as they grow (the baby produce is perfectly edible and gives you a small early crop to eat! or you can space them out a bit more and not worry about thinning them down.
Carrot fly is a problem when you thin them out - especially if not in a raised bed about 10" deep, people like to plant marigold flowers beside their carrots as this smell "hides" the carrots from the carrotfly
Go with a few different varieties - I like a nantes early (so you get an early crop) and I also like the samurai red and there is also a really sweet carrot that I like to (name escapes me)
carrots can be left in the ground until november'ish when the frosts start to come, or you can lift them, store them in clamps or simply, wash, chop, bag and freeze
Sweetcorn, I thought i was gonna get a good crop this year, but just cooled too much to ripen them off - Boooo!!
plant them in a conservatory/greenhosue as seeds and let them get a good start, once they are a foot high, start to harden them off a bit giving them some time outside each day when it's warmer. once it's "summer" put them into a bed, plant them close together so they cross pollinate each other. Then leave them too it - if it's warm and the soil has some moisture in it, they'll do well.
Tatties are easy, buy some seed tatties in the next few weeks, let them chit, plant them just after st paddies day in march. once the seeds are planted, just earth them up - pile earth/compost up covering the leaves everytime they pop through., the highter, the better!!! (though you do have to dig them out at the end!
you can cheat and buy compost bags and just cut a few drainage holes in the bottom and push the seed pot into the bottom of the bag - but that's no fun.
tatties like plenty of water. but best to water in the evening when it's cool - if the soil heats up and water evaporates, you get moisture/humidity and that seems to encourage blight.
Also, you don't have to dig them all up at the same time, once the flowers die off, you can lift a plant at a time, or you can simply cut off all the tops (everything above soil) and then leave the tatties in the ground until you want them - though best to root them all out before frosts start to come and then jsut put them into sacks and store over winter.
I can't grow toms, so you are on your own!!