Where to put my doorbell transformer for new wired doorbell in old house?
Generally, you are controlled by where the existing wiring is located.
Mounting a new transformer has to follow Code and its labeling and instructions, particularly where the AC mains side of the transformer is concerned.  The 24V low-voltage wiring can follow the relaxed rules for low-voltage power, and be installed like thermostat wire.  Here are 2 popular form-factors for 24V transformers:

Mounts to a knockout on the side of a metal box.  AC mains wiring is inside the box, 24V is outside.  Obviously this requires a metal box somewhere whose sides are exposed.
Mounts on the face of a 4x4 metal box, replacing the normal lid.  Again AC mains is on one side, low voltage on the other.

Obviously both of these mounting types require unfinished &quot;utility space&quot; in your home where such things are not an aesthetic misfit.  It's all the rage these days to destroy all utility space in homes in a mad quest for more listable square footage to puff up resale value. Unfortunately those people have painted themselves into a corner, tough luck for them.
You can also simply grab 24 VAC from the furnace transformer, from its R and C terminals (also available at the thermostat sometimes).  It is usually oversized enough it doesn't mind a watt or two for a Ring camera.  Transformers are very tolerant of short term overloads, their limiting factor being internal heat, so the short load of a doorbell chiming won't bother it at all.