You really can use any vegetables at all to make a great vegetable stock. I never make one without onions, celery, carrots, and mushrooms, but beyond that you can change it up to suit your needs and use up whatever you have on hand. Homemade vegetable broth is a great way to salvage any fresh vegetables in your refrigerator or pantry that are past their prime, and you also may want to vary your choices based on what you plan to do with a particular batch of broth.
Onions - Any kind, skins included. If they look dirty, remove the outermost layer of skin or wipe with a damp paper towel. Halve or quarter larger onions.
Celery - Include the leafy green tops. Cut the stalks just enough to fit them in the pot.
Carrots - No need to peel them. Just scrub them and throw them in, or use the ready-to-eat kind and dump in the whole bag.
Mushrooms - These are really good for making a richer, darker broth. I always use at least white button mushrooms in my vegetable broth, and I use portobellos when I want a darker broth, like for French onion soup.
Bell peppers
Parsnips
Spinach
Leeks - Be sure to cut them up and clean them, because they usually have a lot of dirt between the layers.
Tomato - A particularly good addition if you'll be using the broth for something like minestrone soup.
Fennel bulb - Halved or quartered, with greens.
Turnips - Cut in large pieces.
Zucchini
Just about any vegetable scraps - Potato peel, broccoli and cauliflower stems, green bean ends.
Use a very large pot, preferably a stockpot.
Put in just about any vegetables at all (see below). You don't need to chop them unless they won't fit in the pot. In most cases, you don't even have to peel them, because the skins are loaded with flavor and vitamins, and texture will not be an issue when the finished product is strained.
Take some garlic cloves, smack them with the side of a large knife to break them up a bit, and throw them in the pot, peel and all.
Add salt, pepper, and any fresh or dried herbs you choose. I usually use a bunch of fresh Italian flat-leaf parsley.
Pour in one gallon of water.
Heat to boiling, then lower the heat and simmer for about an hour, stirring occasionally.
Remove the pot from the heat and allow the broth to cool a little.
Pour or ladle through a fine strainer, and you're done!
This makes enough broth for two pots of soup, but you can freeze it in smaller portions for other uses.