Yes fertilize! Plants need fertilizer just like we need our vitamins. Here's a little fertilizer 101. Fertilizer has 3 numbers on the front of the package, let's say 16-12-8. The higher the number, the more of that nutrient. The first number (16 in this case) is the nitrogen number. Nitrogen is responsible for green growth (think leaves). The second number (12) is phosphorus. Phosphorus is responsible for flower production. Flowers are important in your vegetable garden too! Without flowers or blossoms your garden would be a flop. Blossoms=peas, tomatoes, squash, etc. The last number (8) is important in cell function. Knowing these numbers makes it easier to choose a fertilizer. If you see a 16-4-4, it would be good to assume that this high nitrogen fertilizer would be good for lawns (where you want lots of green growth), whereas a 4-16-4 would be good in flower beds, or your garden.
Now that you know what the numbers means you can choose which kind of fertilizer you want to use. We're going to break them into three categories. Slow release, granular, and the type you water with (Miracle Grow).
Slow release (Osmocote is a common one) is coated so as you water a little bit of fertilizer is released over a longer period of time. Especially good for hanging baskets because they are watered so often.
Granular is just your common fertilizer that you spread and water in. This is the type you use on your lawns, but it can also be used in flower beds and gardens. It has to be applied more often than the slow release
And then there are the fertilizers that you dissolve and water with (Miracle Grow). These need to be used more frequently than the other two, and can be used in combination with a slow release fertilizer. Best used on flowers and gardens.
It is important to water fertilizer after you apply it (except for the Miracle Grow) because fertilizer sitting on plants can burn the plants. Watering it will cause it to work its way to the soil preventing burn.