I am glad you will start with 'starts' or plants grown from a seed to a decent size.
The least expensive pots are black plastic. You can get them from a nursery very cheaply. Any pot with a drainage hole at the bottom will work. For aesthetics try to keep your pots all the same. You could add a few nicer looking pots but make sure they 'match'. Too many types and colors of pots causes a chaotic look. This is your balcony, an extension of your home/apartment and your personality.
For your tomatoes, plan on at least a 5 gallon size per plant. Garlic is tricky. Have no idea what your amount of sun will be for fall and winter, but garlic is normally planted late summer and harvested the next season. I would go with a shallot myself. I planted 20 in a 'fabric pot' and had incredible success last year.
I would also try potatoes. Those can be done vertically as well. Start in a 10 gallon pot using 3 chunks of a seed potato, install a 'tomato' cage. As the potato grows, once it is at 12" high, fill the cage with straw leaving 6" of green potato plant above the straw. When it grows another foot add more straw leaving 6" above the straw. Pack the straw down quite firmly. Potatoes will grow in this straw. Never allow potatoes to see the sun as this will cause a greening of the potato that is poisonous. If you see a potato pack in more straw to shield it from the sun.
The MOST important thing for you to do is use potting soil from the store. No fertilizer should be added in the potting soil that is in a bag, no moisture holding gels or sponges in that soil...just plain, sterilized potting soil. Nothing else in your pots except for the plants and that soil. No rock or gravel above the drainage hole and beneath the soil. You will probably water every day or every other day. Lift your pots to feel the heft after watering. When they are lighter, water. Do not water if they feel heavy.
You will also have to fertilize. Use Osmocote 14-14-14. You'll only have to do this once maybe twice for your entire season. Follow directions. Do not mess with composts when using pots. Do not add anything else. That is why I like this fertilizer. It is extended release and you can't go wrong unless you try to add manure or mulch or garden soil. Too much fertilizer and you will kill your plants.
Lift the bottoms of your pots off the ground using flat rocks, pieces of broken tile. That is the best thing you can do for drainage. I would also use extra pots turned upside down to change the levels of your garden pots. That will help with light and air circulation and look very cool.
Peppers are very happy in one gallon pots. As soon as they turn red harvest and string and hang to dry unless you know how to roast.
Tomatoes; get the tomatoes off the vine as soon as they turn red to promote even more tomatoes. Allow wasps, bees, flies to visit for pollination!
Try an eggplant! Does well in a one gallon pot. Herbs in a long pot that is at least a foot wide are happy planted altogether and Mint, Oregano, Thyme should last a few years if you don't have very cold winters.
Your only problem and a major one at that, will be daylight hours. If you are going into short days, less than 16 hours of daylight, this will not be a great time to start a vegetable garden. You will need artificial light and that would mean a room inside your home with artificial lights (look up artificial lights on our question/answer history).
You should be able to grow your herbs; thyme, oregano, mint, lettuce, spinach right now, but the rest will not be able to grow without longer day light hours (16 avg). Oh include sugar snap peas to this list!
Here's one idea for balcony gardening.
You'll have time to get soil and pots and pot feet or 1/4" thick tile pieces for lifting the bottom of your pots off the surface they sit, purchased and organized to grow vegetables in your spring. Do you have a neighbor with a balcony right below yours? You'll want to consider how to deal with that. When you water, water should come out of the drain hole and onto your balcony to be hosed off. How will this affect your neighbor below? Do you have any constraints for your development for potted plants on balconies?
I hope this helps. I hope this engenders more questions to ask us!