Check Craigslist! Quite often there's plants listed in the free section, whether it's extra vegetables they didn't need to plant in their garden, flowers they don't like anymore, or even trees that they want dug up and out of their yard. While you're there, look out for free dirt, bark/mulch, and planters!
Another way to get free plants is through propagation. With some varieties you can actually cut off a stem and plant it directly in the soil or let it sit in a glass of water and it will grow roots! You can also pick up rooting hormone, a powder you can drip the stems into to help stimulate root growth, but don't use it on edible plants.
Make sure the plants you want will grow well in your area and take note of the time frame each should be planted. There's no point in putting in all that effort if the plant is going to struggle to survive.
It's all worth it, kids. Old-fashioned, flavorful tomatoes, fresh herbs for cooking, strawberries waiting to be picked... whatever you choose to grow it will be amazing. Just a little bit of care and watching over, and the plants will flourish.