There was a recent thread here about controlling mid-tone contrast, which is virtually the same thing trom a technique perspective as how to control local contrast. You may want to refer to that thread, but here is how I suggest thinking about it.
Assuming the negative contains the information you need, to begin with you control local contrast in any part of the tonal scale primarily with the grade of paper (or filtration with variable contrast papers). Normally a common printing technique is to key the base exposure to important detailed high values or upper midtones, and then adjust contrast to get the shadows where you want them. If everything falls into place, you end up with the right total contrast and local contrast, and only minor burning and dodging are required to fine tune the print.
However there are times when this approach may not result in the desired local contrast in key values, or throughout the print. In that case, modify the procedure to choose the filtration that gives you the desired local contrast. Then, you use burning and dodging to control the highlights and shadows. So in essence you are using the paper grade to determine local contrast first and then using other controls to manage total contrast. Note that to take the print to the "next level", when burning and dodging you might want to add multiple filtration to your toolbox. This means burning and dodging with different filtration (different contrasts). This can help alot in controlling shadow and highlight contrast in addition to helping bring in dense highlights etc. For example you might print your base exposure at a higher contrast, to get good local separations in key midtones, dodge some shadows, then burn in some highlights at a lower contrast. Any combination is possible.
This is where I would start if you want more local contrast. You can develop a lot of technique this way and for the vast majority of prints you won't need more than good, careful multiple filter burning and dodging skills.
Then there are additional controls you can learn such as local chemical bleaching of the print, masking techniques etc. But it is best to start simple and only introduce as much complexity as is required.
Hope this helps.