Our well water tastes very good but is extremely hard (220 mg/L). Do I need to make any adjustments when cooking or baking or will it work fine?
Several things can be affected by hard water. If you're cooking fruit or vegetables in it, the softening is slowed by the dissolved calcium - it reinforces the cross-links in the cell walls. I'm not sure how strong the effect is; if you do boil vegetables, and you find that they get soft enough fast enough, or that you like that they don't soften too much, then I suppose you have nothing to worry about! But for things like dry beans (as FuzzyChef mentioned), where softening is critical, you might have a real problem.
More of a problem might be bread, which is probably the primary baked good you actually use water in - and it makes up an awful lot of the dough. Once again, the dissolved minerals help cross-link, and so you end up with firmer dough. I searched around, and found this reposted article from Bakers Journal about the effects of hard water on baked goods, which says that water above 200 ppm calcium carbonate is not good for breadmaking, but that you can compensate by using extra yeast and adding acid. I don't want to quote the whole thing here, but it does discuss in more detail, so it's worth a look! I would personally be inclined to avoid the issue by using softer water for breadmaking if possible.
The information besides that from the linked article came from On Food and Cooking, a great general food science book. I don't have any personal experience with cooking with hard water, so unfortunately I can't vouch for the completeness of my answer!
The only food I know for certain which will be strongly affected by your hard water is dried beans. At the level of hardness you describe, you may find that they never soften no matter how long you cook them; I've seen reports of up to 12 hours for cooking beans with very hard water. So use soft water for cooking the beans.
There are probably other foods which are chemically affected by hard water, but several Google searches have turned up no general guides on the topic.