... wondering about how to practice a new piece. Do you first practice the piece focusing mainly on getting the notes/fingering down? And then after you are pretty confident with that part work on volume/pedaling/phrasing, etc? When I try to focus on trying to incorporate everything at once I feel like my mind is going to explode.
It is a good aim to avoid brain explosion, I always stress the importance of practice method and really it needs to improve with easy and efficient to learn pieces not always challenging pieces that boggles the mind. The problem I find most people face when initially learning a piece is listening to themselves effectively while playing. If your ear knows very well what something should sound like it will control your actions, however if your actions are so busy that you cannot hear yourself you must stop and reassess.
...it's tough but worth it, it will allow you to 'finish' the piece easier. the notes only method you'll have an easier time starting a piece but a harder time finishing at least that is what i experienced early on before i [begrudgingly] made that paradigm shift in my approach to learning, it's probably one of the reasons may people get to 'intermediate' level and stay there not quite progressing much beyond that, vs those that learn the language of music to a deeper extent and make slow continued progress to better and better musicianship and more advanced repertoire....