is this even possible given my scenario?
Yes, if you select the proper veggies.
what vegetables (or fruit) would I be able to grow successfully?
Most fruit and veggies need full sun of 6 hours or more. You would need to have some idea of how long the sun is on your balcony. There are some veggies, and a few herbs, that will grow in part sun of 4 to 6 hours. Here's the link that explains it all.
Here are some that come to mind that can grow in part sun. Some of these prefer full sun in spring and fall but need part sun in the heat of summer.
arugula, beets, Chinese cabbage, celery and celeriac but need lots of water and not for the beginner, eggplant, Egyptian onion (only onion I know to tolerate less then 10 to 12 hours of sun), kale, leeks, lettuce, mizuna, mustard greens, orach, pak choy aka bok choy, radicchio, radish, rutabaga, shallots, spinach, tomatoes need at least 8 hours of direct sun, turnip.
Broccoli and cauliflower can tolerate some shade but will be slow to mature.
Herbs that can tolerate part shade -
chives, tarragon, sweet woodruff (great for May wine).
when would I start to plant?
If you are going to grow from seed, they are started indoors about 6 weeks before the last expected frost date for your area. Then the small plants can be planted outdoors. The seed packets will tell you when to plant the seeds indoors.
If you will be purchasing starts (small plants) they can be planted outdoors after the date of your last expected frost.
Here's a frost chart for Canada.
what would I need to get started (please be specific)?
You will need some type of scoop or trowel. If you will be planting seeds you will probably need some type of tweezers or small scissors like the type used to cut baby's nails. Pots of different sizes, depending on what you want to grow. Seed starting kit of you want to grow from seed. Potting soil, preferably organic, for the plants and seed starting mix if you plan on growing from seed. A watering can or hose, preferably with a sprinkler nozzle and organic fertilizer. Fish emulsion mixed with seaweed is great.
Here's some helpful info on growing veggies in containers.
You might also find these sites about starting seeds helpful.
When to harvest veggies:
That should get you started.