Herbs will grow best in a sunny, south-facing window. For optimal growth, herbs require 6 hours of sunlight each day, or you can use fluorescent lights, keeping them on 14 hours a day, 3" above the plants.
Start with 4" or 6" containers (depending on your windowsill or shelf width) that have drainage holes in the bottom. Fill them with a high quality potting soil that has good drainage. An option is to fill the top 1" of the container with seed starting soil which is less likely to contain fungus or disease pathogens. Thoroughly moisten the soil by sprinkling with warm water or preferably, by setting the containers in a tray of water, so the water will be absorbed from the bottom.
Once the soil is thoroughly moist, sow the seeds according to directions on each packet, then cover the containers with clear plastic to retain consistent moisture. After the seedlings have emerged, remove the plastic.
Once plants are established, err on the side of letting the pots get a little dry between waterings.
If you used a potting soil that contained fertilizer, your herbs may be just fine without any additional fertilizer for months. Herbs have the best flavor when grown 'mean and lean', so they don't need a lot of pampering. If they do start to get a bit pale, you can add a light dilution of fertilizer every 2 weeks or so.
Growth and Care
The growth of indoor herbs may be a little slower indoors than outdoors, which you may enjoy if you just like a little snip now and then for some fresh flavor in the kitchen. You can also trim the plants from time to time to maintain a compact shape. Snip off any flowers that appear; when herbs put energy into producing flowers, they may become less flavorful, or bitter. If the herb stems become woody (i.e. basil, sage, thyme), it's time to transition them outdoors (if weather is warm enough) into a larger container, or the ground, or toss them in the compost pile and start fresh plants from seed.
Some herbs for a long-lasting indoor herb garden include: basil, chives, cress, garlic chives, lavender, lemon balm, marjoram, mint, oregano, rosemary, sage, shiso, thyme, watercress. You can grow these varieties indoors for months, harvesting leaves as needed.
Herbs that would be happier outdoors (due to long taproot or desired maximum growth for flower production) include: Borage, chamomile, clary sage, cilantro, chervil, fennel, feverfew, lovage, and parsley.