Oh boy AJ, you came to the right people. Yes you can grow things all year round in LA, though plants will slow down in the winter, it's just how they're programmed. Sometimes it will freeze here at night, so you want to bring delicate plants (like Basil) inside. When we had a kitchen window that got enough light, we grew herbs just inside, and it sounds like you can do this too.
Here are some we recommend (we're growing a bunch of these ourselves):
• Basil (a must!)
You can either plant them all in one pot (though that would be a large pot and the mint would most likely try to take over) or, our choice, plant them in indivual pots and line the window, or shelf or wherever you're putting them. We normally buy herbs at a local nursery (we bought our basil at trader joe's and haven't had a problem) as small plants in 2-4 inch pots and then replant them into something a bit larger and prettier (we love ecopots for their great colors). When you water them (about once a week) make sure not to drown them, it will make the roots rot, so find that balance between too much water and not quite enough, the plants will tell you.
[image from gardenerresource]