There's definitely an art to processing Milky Way shots. First, shot the core. It rises about midnight right now in the northern hemisphere. Make sure it's properly exposed (base exposure is 30s f2.8 ISO 3200). Adjust exposure and contrast as needed in Lightroom. Boost vibrance and saturation as needed. Ensure proper white balance, for my 6D that's usually about 4000K with +6 tint. But white balance is subjective and will depend on location due to light pollution.
There are some great tutorials on Youtube from photographers such as Ian Norman (Lonely Speck), Michael Shainblum, and Chris Ewen Crosby Photography.
For adjusting white balance I use the technique that Ian Norman mentioned in a video - bump vibrance and saturation to 100 and then adjust temperature and tint for proper balance. First adjust the temperature until there's a balance between blue and yellow in the image, then adjust tint until there's a balance between magenta and green. You might have to go back and adjust temperature again after adjusting tint. If you prefer an image that's more blue then adjust the temperature down slightly. Once you have the temperature and tint balanced reset the vibrance and saturation to zero. If you have a neutral white balance the image should appear to be black and white. Now you can boost the vibrance and saturation to a pleasing level.