I had great luck with window boxes of parsley when I lived in Germany. Parsley is very forgiving, just water it when it looks wilted.
Last year’s garden had huge (4’ tall) basil plants. We bought the starters at the local garden center and they just take off. They are very forgiving also, water when they look wilty. Basil will really take off if planted where it is sunny.
Rhubarb grows very well with little care. You don’t get much to harvest until the second year though. It is also prone to rot and mildew diseases.
I have wonderful horsradish also. It grows like a week so plant it in a bottomless bucket so it does not take over the world. It will come back each year, whether you want it to or not.
Dill grows easily, I don’t grow it since I don’t cook with it.
Sage is beautiful and grows easily. Chives are a perrenial (like many herbs) and are hard to stop once they start growing.
To tell the truth, most of the common herbs will grow with very little care, this makes sense since they really are weeds, but useful weeds. I live in eastern South Dakota, we generally get enough rain that I don’t have to water much, usually the only concern I have is critters (bad bunny, naughty squirrel) and weeds.
If you can find a copy of Rhodale’s Herb Guide it is the standard reference for herbs, it gives all sorts of information; culinary, medicinal, horticultural and for dyes.
You can also visit your local Agricultural extension office for more information on gardening. The extension agents are a wealth of knowledge for help and identifying the lothesome creatures that will try to eat your bounty. The USDA website has information that will prove helpful.
Herb gardening is perhaps the easiest gardening, takes little space and the herbs make great gifts. By the way, your SO will probably roll his/her eyes when you use crushed mint leaves on your forhead to feel cooler on hot days. I can’t say I blame them but it does feel really good.