you need 110 volts on one side of xfmr, which will be installed through one of the knockouts in your panel. On the other side of the xfmr will be where your 12 volt wire connects, then run that wire any way you want to the doorbell chime.
I've always liked the wireless ones myself, no fuss
Depending on the mounting provided, you might find it easier to attach the transformer to a junction box rather than directly to the panel. You can install it anywhere that is accessible and where you have 120 volt power although there aren't many places you can install it without its being unsightly. Junction boxes can be attached to panels using nipples as opposed to wired in with a short length of Romex.
When cleaning anything including baby diapers, it is better to sweep off or scrape off as much solid and semisolid material as you can first. (1) Less cleaning fluid or detergent needs to be used, (2) Less of a mushy mess which in turn means that bleaches or disinfectants will be more effective.