A bare-bones approach is to use an incident light meter and then increase exposure by about one stop to better catch the shadow detail (assuming you are using negative film). For difficult lighting, you can bracket exposures around what you think is the correct one. With roll film, this could be the best choice, especially because of what LJH says above.
Once you get used to the film(s) and developer(s) with this simple exposure system, you can more easily work-in the zone system. You can also use a given film in smaller format (135, 120) to get used to it and save money before moving to larger format (4 by 5 etc.) film if that is your plan.
It is also worth knowing the shutter's accuracy and adjusting accordingly, especially for mechanical shutters on old cameras.
Modern films have more exposure latitude and thinner emulsions than what Adams typically used (although he does consider the more modern films of his time in "The Negative"), so some of his techniques are not as necessary sometimes. To me anyway, Adams was very keen on exposure, and his guidance on this is very clear and valuable, but perhaps the zone system is more complicated than it needs to be to start out.
I don't post here much, but have been using B & W film since the late-1960's and slide film since about 1970. The newer films do seem more forgiving regarding exposure.
Good luck and have fun!