Re: And WHERE do the moodle backup files go?

I highly advise that you look at getting a vps - at least make sure that the new host can support the database requirements - barracuda file format and the following mysql settings:
[client]
default-character-set = utf8mb4

[mysqld]
innodb_file_format = Barracuda
innodb_file_per_table = 1
innodb_large_prefix

character-set-server = utf8mb4
collation-server = utf8mb4_unicode_ci
skip-character-set-client-handshake

[mysql]
default-character-set = utf8mb4

Note that a couple of these are global variables and there are very shared hosting plans that support them...