I have two databases in MySQL.  One has all the data I need but it is
all in varchar format.  Another has the table structure I want, but is
empty.  I have tried using phpMyAdmin to export just the data from the
first table into the second but I get an error when the program tries
to insert data into the first non-varchar part of the second table (an
enum field).  The question: how do I move the data in an all varchar
database to one that has other field types (varchar, enum, tinyint)
automatically

A two step process may work for you.  First export your data to a
plain old ASCII text file (which has no typed variables).  Next,
import the data from the temporary ASCII text file into the other SQL
database with your desired table structure.

I would also try creating a backup copy of the database WITH the data,
and just changing the vartypes on all of the fields.  You may end up
losing some data, but it might do the trick.

Short of that, you could script it with something like PHP and let the
PHP script all of the data tye conversion to make the structured
database swallow the mis-categorized data.