I need someone to find out how long it takes for the average renter
to find a place to rent, and also how much, on average, they spend
finding a good place to rent.

I would say it takes on average 20-30 days. Reasoning is folks think
about moving, kinda look around, firm up their decision to move, give
their "20 day notice" to move, then have 20 days to vacate. A rental
owner is not going to hold an apartment or rental house open for more
than a couple weeks unless it needs considerable work, therefore, you
won't find a landlord where you can find the rental and have a
landlord hold it for longer than say 20 days or so, UNLESS they have a
lot of vacancies. If the rental is ready, they want it rented ASAP..

It is also affected by where you are and what season it is. Nweport,
RI in July is impossible while West Palm Beach, FL at the same time is
pretty easy.

Thank you for your help.  I appreciate it greatly.