I quite understand. There is a learning curve to using MySQL for sure.
I would use digitalocean.com servers, preloaded with a MySQL image (one click deploy), but there are free ones to try out, such as https://www.freemysqlhosting.net. Fairly basic but it does work.
You would then need to ask the Anvil guys to add a mysql package (I use “pymysql”) to your account so you can import it in your code. You would then access it like this :
import pymysql.cursors
...
connection = pymysql.connect(host='x.x.x.x',
user='xxxxxxx',
password='xxxxxxx',
port=3306,
db='mydatabasename',
cursorclass=pymysql.cursors.DictCursor)
try:
with connection.cursor() as cursor:
sql = "SELECT * from atable WHERE tablefield=%s"
cursor.execute(sql, ('searchterm'))
result = cursor.fetchone()
print(result)
finally:
connection.close()
There’s a slightly better example here : Examples — PyMySQL 0.7.2 documentation
EDIT -
you could also use something like mongodb (a NoSQL database). You can use it for free here mlab.com. You can connect using mongo python drivers (assuming Anvil will load them for you) or a straight forward REST API. Pretty well documented.
Difference? NoSQL is not SQL, ie it doesn’t use the SQL language and has different ways of searching for data. Tomes written on this subject that google will help with, but it may or may not suit your use.