Aphids - A jet of water from hose pipes will knock them off, and thankfully they cannot climb back up, but where this is not possible, dilute a few drops of washing up liquid in water and use a spray bottle. Target the undersides of leaves as that is where they'll be sheltering.
You can do the same with whitefly though regular doses will be necessary. Whitefly populations under brassica leaves (swede, brussel sprouts, broccoli etc) can get horribly out of hand, especially in late summer/autumn if they have been allowed to flourish uninhibited. Try companion planting - something like french marigolds between crops can sometimes help.
Caterpillars - agree about the netting. The size of holes is critical - the finer the netting, the better. I've seen cabbage white butterflies squeeze through the tiniest of holes, and they seem intelligent enough to see that tiny gap you thought they'd not notice. Also, make sure the net is not touching the crop as the butterfly does not need to land on the leaf to lay eggs. You should still inspect crops for tell tale signs of eggs or caterpillar damage and dispose of them.
Cabbage beetle - would that be flea beetle? I tend to find they do most of the damage on the outer leaves especially in spring on young plants (radish and turnip in particular), but once they start to establish, the heads of cabbages that develop are usually fine. I've never been too bothered to feel I need to take drastic action, and I don't expect my crops to look as pristine as pesticide loaded supermarket ones
Take the holey outer leaves off, and no one will know any different.
Slugs - as Yorkie says. During the day, slugs and snails tend to hide under rocks and wood, so see what's in the garden and have regular hunts. Flinging them over your neighbour's fence (
did I really say that out loud?) won't get rid of them - it's been scientifically proven that they WILL come back! The older version slug pellets contain metaldehyde - you really don't need to put too many pellets down (it is a bait which they eat). A light sprinkling is all you need, and try not to put them too close to your plants as this would be like inviting them to their food source instead. The risk to wildlife (birds and hedgehogs) is if they eat the ones that have ingested the pellet, so removing the dead slugs would be a good idea. If you have crops under netting this won't be such a problem.
Potatoes - perhaps you were lucky or maybe conditions were dry, but slugs also like to burrow into potatoes. I would recommend you choose varieties that are slug resistant (better still if it is also eelworm resistant as sometimes it can be difficult to know which culprit is responsible - one may just be taking advantage of the holes made by the other)