Herbs that do NOT do well indoors include sweet basil, lemon verbena, summer savory, and tarragon. These are all poor choices because they either go dormant or shed their leaves excessively. Most other herbs do very well in a south-facing windowsill. Oregano, thyme, parsley, and sage can all be grown in small pots and trimmed as needed for the kitchen. Pots of rosemary and sweet bay (Laurus nobilis) are equally valuable, though they need more space. In the onion family, windowsill candidates include chives, onion sets, and garlic cloves. The onions and garlic can be started in pots. Clip emerging foliage when it reaches a couple of inches high. If you have only an east- or west-facing windowsill, try mints such as peppermint, spearmint, and lemon balm. Provided they are sheared regularly, these, too, make excellent houseplants.
If you find you have insect pests, remember that plants intended for human consumption cannot be sprayed with the pesticides used on ornamentals. Discard infested plants.