Growing herbs indoors
It's easier if you know some botany.
There's a few things to remember when growing herbs indoors.
1) Plants in the Umbelliferae (Apiaceae) (carrot family) have taproots.
So don't grow dill, parsley, chervil, caraway, coriander, sweet cicely, aniseed or lovage, unless you can give their roots lots of deep room. Buckets are deep enough, but they're just a teensy bit less than elegant on the windowsill.
You can grow caraway in a normal pot, but the plants will be dinky little things, not robust healthy ones from which you can coppice enough leaves for a caraway leaf soup.
Don't even think about growing angelica indoors. It's huge, large, simply enormous, and the seeds sprout in the same autumn they are produced - the germination of storebought seeds is rather too close to 0. Our angelica also requires a winter.
(Horseradish, too, has a taproot; that's not an umbellifer, though.)
2) A lot of the culinary herbs in the Lamiaceae (mint family) are from the mountains north of the mediterranean. That's some very porous dirt.
And that means that rosemary, lemon balm, lavender, hyssop, marjoram, oregano, savory, thyme and sage all like their feet more or less dry.
The extreme example of this is rosemary, which is evergreen - until it dies, that is. So you water it once a week, no problem, except that rosemary really does need to dry out totally and completely between waterings. If it hasn't, its roots rot. Your plant will still look healthy for a while - it's an evergreen - but it is in fact dead. So your "but I haven't watered it for three weeks now!" came too late for your now late plant.
Exceptions to the "dry feet for the culinary Lamiaceae" are the mints (that's Mentha spp.) (they love water) and basil (ditto).
3) Annuals are easy.
That means that basil, watercress, summer savory, coriander, dill and chervil all sprout easily and are happy with whatever you give them - up to a point, for the umbellifers.
4) Biennials and some annuals will turn into short-lived perennials if you keep them from flowering.
I've done that with basil and a few others.
5) Don't worry if one or the other plant dies - that just means that you have room for another plant now.
Other plants I've grown successfully on a windowsill? French tarragon, which will look dead dead dead in spring, but which actually is just taking its time coming up again. And garlic, which I've grown for the greens - for bulbs you'd need one bucket for each garlic you're growing.
There are a few more tips in the culinary herbfaq.