Look into square foot gardening. It's basically gardening in 4 foot by 4 foot by minimum six inch boxes divided into sixteen square feet. It is supposed to use Mel's mix (Mel being the guy who wrote the book) which is 1/3 peat moss, 1/3 compost, and 1/3 vermiculite and can work out to less money than traditional potting mix (but not always, depends on the price of potting soil and the price of components). There is a book just titled Square Foot Gardening which your library might have and a lot of information online which can be useful to small gardening even if you don't go with that method .
Also check your library for books on gardening specific to your area, and look online for garden bloggers or forums. I live in a very cold climate now, but I grew up in Oregon, and even there my mother was able to garden pretty much year round. In the winter it would mostly be kale and other hardier greens.
There are lot of cheap/free containers available if you get creative. Try freecycle or thrift shops for old rubbermaid storage bins (the shorter kined work better) and old baby bathtubs with flat bottoms, just punch some holes in the bottom for drainage. You can grow some things in the large size of coffee cans. There are instructions online for growing verticle gardens in pallets that have been stood upright, or some people garden in old tires. Greens and many other things do just fine in pretty shallow containers (though you may need to water more often). For tomatoes and things that need larger ones, five gallon buckets, which you can often get for free, work well.
Salad mixes can be a pain because when stuff is first coming up it can be hard to tell what is a weed and what is mean to be there when the seedlings all look different. I prefer to plant just one variety of lettuce/green for each container or area so that the seedlings all look the same.
Strawberries are my kids' favorites to grow. You can get seeds on Ebay pretty cheaply for unusual varieties like bright red or purple carrots, golden beets (I prefer them just for the stain factor; just as yummy as the red ones, but without the ruined clothing), and bright purple cauliflower.