Hi Emilie. I use lemon oil sparingly (once a year) to clean the fret board. I use Stelling Glyde-cote on the woodwork (keeps the banjo smelling like new) and Simichrome polish on the metal parts used very sparingly.
Caution - Do not use Brasso or other abrasive metal polishes as it is likely to wear off the electroplating.
To be honest I don’t clean my banjo all that often save for wiping the strings down with tissue after playing. I stopped using Fast Fret years ago as it seemed to leave dirt deposits on the banjo head. Maybe I just wasn’t cleaning off the surplus properly anyway my strings feel fine without it. I use a pencil eraser to clean fingermarks off the head. I keep my banjos in their casers when not in use that way they don’t get very dirty.
Light gun oil on the threads of the head tensioners (one tiny spot on each) just to keep the threads from seizing between head changes. Especially important If you live in humid climate to prevent rust - metal oxidisation.
Tools, Soft clothes - one for wood one for metal polish. small paint brush and pipe cleaners to brush the dust from the pot, pipe cleaners get those hard to reach nooks and crannies.
(Ask @BanjoBen what he uses to clean off the mouth drools - I don’t have that problem so I have never had to deep clean the pot wells yuk )
I guess today I’ll be cleaning my banjos, since you reminded me it’s been a while.
P.S. I always wash my hands with hot soapy water before I pick up my banjo. which reduces the sweat and oils from my skin getting onto the banjo.