All you need to grow fresh, nutritious vegetables any time of year is a little bit of window space, and some sunlight. Window gardens work best with a south facing window (in the northern hemisphere) to catch the most sunlight, but work almost as well with an east or west facing window.
To start a window garden, get any simple container, plastic or ceramic. Make sure that it has decent drainage– you don’t want your garden to get too wet. Add some soil (available at any garden shop or available here). Eventually, you will want to add some compost (no fertilizer necessary!), and the best kind of compost comes from worm castings that you can make in your own home. Bury your seeds in a quarter to a half inch of soil/compost depending on how big they are (bigger seeds should go deeper). Water well so the soil is damp to the touch but not soaking wet. Just check back in every day to make sure the soil is not dry. If it is, just water a bit.
Here is a picture I took of a small window garden in my apartment. 
The big plants are Basil, which have been growing there in that window for OVER A YEAR! I snip off a bit whenever I want a nice fresh flavor in my juices or salads. The smaller plants are a combination of Arugula, Cilantro, Parsley, Tatsoi, and Rosemary about 3 weeks after planting. Sure, they are probably a little too close together and might not grow up to be the biggest plants, but it doesn’t matter. Chances are some of them won’t survive, and I can always thin the smaller ones out or dig them up and transplant them.
The number one rule in gardening (and probably the only one you’ll need for years after you get started) is “PUT THE #@?%! SEEDS INTO THE GROUND”. You will pick up most of the other stuff yourself, or from more experienced gardeners as you go. You’ll really be surprised how successful you can be following rule number one.
Recommendations
Organic Basil Seeds
Organic Arugula Seeds
Cilantro Seeds
Parsley Seeds
Tatsoi Seeds
Rosemary Seeds
Worm Factory 360 Worm Composter
1000 Red Wriggler Worms