A Simple beginning...

Vegetable broth is a staple in most of my cooking.  I try to plan ahead and throw together broth while in the kitchen making other dishes just so I have it on hand when I need it.  It takes only a few minutes to trim the veggies and throw them in the pot with some water to cook.  The only challenge to making any broth is maintaining a steady low simmer the duration of the cooking time, no longer than 2 hours.  I seem to recall reading that vegetables cooked longer than two hours add no appreciable flavor to the broth, so I aim for two hours, but have made decent broth in just under an hour while assembling all my other ingredients.
I don’t make large batches of broth and freeze them, so I can create the flavor profile I am looking for in the dish when assembling the broth.  Many vegetable and herb additions will give a decidedly ethnic flare to the broth and will enhance the overall dish, but may not be welcome in every recipe.  However most broths, vegetarian or meat, start out with the basic trinity and expands from there.
Basic Vegetable Broth
3-4 large carrots ends trimmed, cut up
2-3 stalks celery, large chop
2 onions skins removed, cut up
6-8 cups cold water
Place ingredients in a stock pot adding enough water to cover the vegetables.  Bring to a boil slowly, skimming off any foamy impurities that might rise to the top of the cooking broth.  Reduce the heat to a simmer and let cook for 1½  to 2 hours.  Strain through a fine mesh strainer, pressing cooked vegetables to extract all the juices.  
All homemade broths will need salt but add it sparingly prior to final tasting of the dish since any number of additional ingredients may contribute saltiness.
Other flavor options:
Oriental Vegetable Broth:  Add an additional 1-2 shallots, 5-6 scallions (green onions), 1 2-inch piece of ginger, 2 cloves garlic
Italian Vegetable Broth:  Add 2 leeks, 1 bunch flat-leaf parsley, leaves and stems; 1 bay leaf, 1 fennel bulb sliced (optional),  1 sprig fresh thyme (optional)

Mexican Vegetable Broth:  Add 2 cloves garlic, the stems and discarded pieces of red pepper if available, and dried ancho, poblano, or other peppers.  The dried peppers available at Mexican grocery stores add significant flavor and need little more than to be cooked in the broth.  The flesh can then scraped into the dish for an added flavor boost.