I also buy supermarket pots and normally pull the separate plants apart then transplant them, the chives do particularly well in the garden so I no longer need to buy any as they come up every year.
This year I have no compost so have left them in the pots inside slightly taller glass jars (IKEA vases) so that the herbs don't fall over. Apparently you're supposed to water from the bottom but not leave them sitting in water so, before they completely dry out, I water into the vase and then tip out the water after a few minutes soaking time.
There are so many little individual seedlings in the pots that they pretty quickly exhaust the food in the compost but most herbs like a fairly poor soil, except perhaps for the lusher basil plants, so I just add a very small dash of plant food to the watering can once a week. Personally I nip off any flower buds, but you might want to use flowers in your salads.
I've found that the "Greek" basil is the longest lasting, its a pretty little bush with smaller leaves than the normal version, the flavour isn't quite the same but I drop a sprig in my cooking and can't really tell the difference.
I've got a west facing windowsill so they get lots of light, but will have to move them back into the room later in the year when they start to get direct sun in the afternoon.
I never have much luck with coriander and would appreciate some tips on how to keep it going. It seems to grow on single stems and doesn't come back after you've snipped the leaves.