This is indeed a pretty broad question - it's essentially my whole field of research plus a lot of extras. But I'll address the specifics, and try to give an idea of the big picture.

So, turnips are very, very far from being the only English grown food still eaten. That's a weird one to pick, to be honest, since turnips are so very little eaten now in comparison. Wheat. Barley. Carrots. Cabbage. All of those were grown in the medieval era and are more widely eaten in the modern era than turnips. Tomatoes did, though, only become part of Italian food after 1492 (and quite a bit after, in most cases; they were considered poisonous for some time and only grown as ornamental plants), and the same is true for potatoes in Ireland.

We have a fair idea of what food was like before the Columbian exchange in some areas, though not all. This comes from collections of recipes, from Roman times forward, from archaeological findings (and particularly the relatively new fields of archaeozoology and archaeobotany), and from evidence in paintings, murals, and literature. In, say, 15th century England, we know a lot of what was eaten, because we have lots of evidence from all three areas. In, say, 7th century Norway, we have no recipes, no paintings or literature, and not a huge amount of archaeological evidence.

We know more about the food the nobility ate - in most cases a lot more - because almost by definition, any written sources are connected with educated, and therefore richer people. There are plenty of areas where we know in considerable detail what the aristocracy ate, while the peasant diet is a guess of 'bread, beer, stew, and leftovers'.

In Ireland, my area of particular study, medieval food featured a lot of dairy - cheese and butter, specifically - and then grains in the form of either porridges and gruels, or bread, or beer. Onions and wild garlic would have made up a lot of the vegetable portion of things, and various other 'herbs' as well. Translation issues from older documents make it unclear what many of these were; we can translate them as the "peasant's herb", "the cattle owner's herb", and "the king's herb" - but we don't know what the actual plants are. Meat was eaten, certainly - pork, and chicken and beef as well as venison and various other forms of game, and seabirds. Fish was eaten, but not nearly as widely as you might expect on an island of this size. Although it's possible that some of that is due to the sources we have - a lot of the information comes from law texts concerned with land and farming, but the law text dealing with the sea has been lost somewhere along the way. And there was honey, and thereby mead.

There were strict laws of hospitality, and who was entitled to what food, and so forth, which give an idea of what the nobility ate, and also, rather unusually in the context of medieval texts, what the peasantry ate. And we have some surviving food - a fragment of burnt bread which came up in an archaeological dig, some seeds and other bits from Viking and Irish sites, and several examples of 'bog butter' - literally, butter buried in the anaerobic environments of peat bogs, which survives for, literally, centuries.

I hope that outlines some of what you're looking for, and feel free to ask about further areas - I know a bit about medieval English food as well, and a bit about 10th century Arabic food as well.