A short growing season is often the biggest reason to want to grow indoor vegetable gardens, but it's not the only reason. Growing vegetables indoors allows you to control more of your plants' environment, keep hungry critters away and be able to eat fresh homegrown vegetables even in the dead of winter. However, keep in mind that you are solely responsible for your plants' welfare and can no longer rely on a sunny or rainy day to supply your plant's needs.
Select a garden location at a south facing window if possible. Although your plants are getting sunlight in the winter, they won't be receiving the same number of hours of light as in the summer, so using a grow light may be necessary even with a sunny window.
Choose a grow light which extends over the surface of the indoor garden. A clamping desk lamp with a small grow light screwed into it works well for smaller gardens or choose longer lights for growing a lot of vegetables. While commercial grow light systems are available, typically a florescent base with grow bulbs works fine and costs less.
Select the type of pots you are used to and that are large enough to support the eventual size of your plant. Standard plastic containers can work well, but they don't drain well and if you overwater, your plants may drown. Clay pots on the other hand will absorb water, and could let the roots dry out.
Water the vegetables regularly and keep track of how long it takes for the soil to dry out; for the health of the plants, do not let the soil get this dry.
Check on your plants twice a day, when you turn your lights on and when you turn them off. This will give you the best idea for how quickly your plants consume their water and how often you'll need to add more. Also check for new growth and pests during this time.
Set up a small oscillating fan if you notice your light bulbs giving off a lot of excess heat. The fan shouldn't be set on high, but should be aimed to flow over the plants to keep air moving and can encourage stronger plants.
Fertilize or apply pesticides as needed. Track your maintenance routines so you know when you can safely apply fertilizer or pesticides again.