This isn't the easiest question to answer, because we're very short on records for kitchen gardens. Such gardens weren't subject to the same laws as field crops - they weren't tithed, and they didn't form part of anyone's rent or duty. And crops grown at this small scale don't leave the archaeological marks that we can find for larger fields - plough marks, pieces of broken implements, and so forth. We do have some records of such gardens in literature and in imagery, though these are obviously biased toward the wealthy and the religious orders.

So with that caveat out of the way, let me first tell you what they didn't have: potatoes or tomatoes, most kinds of peppers, or the bigger pumpkins and squashes. These (particularly potatoes) are a staple of the modern kitchen garden, so they make quite a difference in what's grown.

What would, then, have been grown were: onions, garlic, carrots (in a wider variety of colours than we have), parsnips, turnips, skirrits (a vegetable vaguely like a bunch of skinny parsnips, not much cultivated any more^(1)), radishes, lettuces of various kinds, rocket and various other herbs, sorrel, peas and beans, and brassicas. I say 'brassicas', rather than listing off the particular kinds, because it's not clear to us what kinds there were. Cabbages, certainly, but also sprouts, broccoli, cauliflower, and others in between those. The further north, the more root vegetables were grown; the further south, more leaves and herbs (this is a horrible generalisation, but it's just about defensible).

Peas and beans would have been very important crops, because they dry well and therefore can be preserved through the winter, provide a level of protein that other crops don't, and aid in the longer-term cropping efforts by returning nitrogen to the soil. That's not to say that that would have been understood in scientific terms, but in small spaces, it would have been an immediate consequence.

Fruit trees might have been grown, if there was enough space and although grafting as a technique has been around for a very long time, dwarfing rootstocks are a fairly new development. So apples, pears, etc, would have to be extensively pruned, and probably trained against walls, in order to have them take up less space and not grow out of reach. Soft fruit seems to be more debatable; certainly the strawberry we know is a more modern fruit, and the medieval strawberry was almost certainly the wild variety - small and intensely sweet. Raspberries and blackberries would also have been more wild than cultivated, but currants were grown, and in places further south, grapes in large quantities. Grapes were also grown in places unexpectedly far north when the climate allowed for it - England and the Hanseatic cities, even.

In among these, you can expect that people kept chickens or ducks, and probably pigeons in dovecotes, as well as pigs. Even in a town, the odd goat is not unlikely.

If you need more detail on any of that, let me know, and I'll do my best to follow up.

Sources:

Vera. K. Ninez, Household Gardens: Theoretical Considerations on an Old Survival Strategy (1984)

Mudge, K., Janick, J., Scofield, S. and Goldschmidt, E. E. 'A History of Grafting', in Horticultural Reviews, Volume 35 (2009)

Amandine Cornille, et al, New Insight into the History of Domesticated Apple: Secondary Contribution of the European Wild Apple to the Genome of Cultivated Varieties

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^(1) Although seeds are available, and I do intend to have a go at growing them myself in the next few years.