Buy aquaglobes! Seriously, I always forget to water things, but our basil's lasted through January in our apartment.

As long as you have some direct sun (mine has morning sun) you should be able to grow most herbs. I have a small rectangular pot in my window with parsley, rosemary and thyme and it's doing great. Just keep the soil moist and you'll have fresh herbs all year.

I know it's not traditional indoor gardening, but I have had tremendous luck with Aerogardens. I've never had a green thumb, but the plants in my aerogarden are thriving. And you don't need to worry about sunlight because they have a bulb on a timer. And there are tons of plants to choose from (herbs, flowers, and veggies). I am in no way affiliated with the Aerogarden company, I just really like their product.

Check for pests on a regular basis. Aphids, whiteflies, spider mites and thrips can get out of hand quickly. Spray the plants with soapy water and then rinse after a few minutes. Repeat once a week until you don't see any signs of pests. My shower was full of plants getting de-bugged yesterday!

Rosemary is a very hardy plant to grow indoors. I've never been able to keep Basil alive, it always makes it though winter with one leaf, then dies in the spring!
I've had luck with oregano also, but wasn't impressed with the flavor when I tried to cook with it.

I always have basil growing, it is easy to keep going, you just need to keep it watered, but not waterlogged, in a sunny position. I keep mine in the living room and close the curtains behind them at night so they don't get too cold. Also if you cut the biggest leaves from them every now and then put them in a sandwich bag in the freezer you will have a ready supply for Pesto or whatever. I manage to keep mine going for over a year, though toward the end the stems get a bit wood and the leaves start to loose their vibrant green. I guess if I feed them fertiliser that would solve that, but I just tend to ditch them. It is worth all the effort as Basil is quite expensive here in Tokyo.

If you don't have a window, or find the plants aren't thriving to your satisfaction (i.e. legging, scrawny, bending over towards the window too far) a CFL bulb (spiral shaped fluorescent light bulbs) in a gooseneck lamp over your pot should do the trick for lighting. Make sure you get it close to the tops of plants, at least 6" or so.

Mint is so easy to grow. I have a pot of cat nip, not even really close to a window. I've kept cilantro pretty long too (it bolts outside in the summer, and lasts way longer indoors).

As for tips- use fertilizer! It makes such a big difference. You can get these little pellets that you put in there once and you're good to go- fuller, darker and more dense plants.

Also, don't over water you could end up with little flies living in the dirt of your pots.

Pinch off dying/dead/weak parts of your plants so they can put their energy into growing the healthy parts. Also pinch off flowers/flower buds, these will change the flavor of the herbs.

watch for dryness. I have radiator heating, and for the sake of my herbs (and mushrooms) I've put spare containers full of water in every room on every radiator. I use shower-warming water to fill the containers to keep the air humid enough for the plants (as a side effect, I feel a whole lot better than I did last winter), but I still have to spray my herbs.