One of the best effects to use for this is Color Curves (if Vegas Studio has it). Add the Color Curves effect to the video clip if you want to brighten just one clip, or to the whole track if you need to brighten everything on that track. In the Color Curves window you'll see a graph with a diagonal line going from dark in the lower left corner to light in the upper right. Drag the dot on the middle of the line upwards or leftwards to brighten the middle range of the image. If you want to brighten just the shadows, add a dot near the bottom of the line and drag that dot up.

If you don't have Color Curves you can Levels and experiment first the the Gamma slider.

If you don't have Levels then use Contrast & Brightness. Slight the brightness control up first to get the general amount of brightening you want. The problem with this effect is that you can wash out the highlights and lose the blacks very easily. The Color Curves effect maintains the hightlights and shadows while you adjust the midtones.