Ugh! 'one click installer' ... evidently it didn't follow moodle's official docs for installing!
See: https://docs.moodle.org/311/en/Installing_Moodle
https://docs.moodle.org/311/en/Installing_Moodle#Create_the_.28moodledata.29_data_directory
Guess ...
SiteGround actually has their infrastructure on Google.
SG also has it's own panel, but ... is the one click installer
called Softaculous?
Questions ... because you shared paths in the errors you have seen we know it's Linux ... but that OS? Got ssh into your server? If so, ssh and issue: uname -an
If you are running apache server, one could put an .htaccess file
in your moodledata that contains:
deny from all
AllowOverride None
Note: this file is broken intentionally, we do not want anybody to undo it in subdirectory!
nginx doesn't use .htaccess files.
Suggestion/guess:
lms.breakthroughmathematics.org/public_html/*moodle codegoes here moodledata could be in lms.breakthroughmathematics.org/ at the same level as public_html
The config.php file in moodle code has to have the path to moodledata. Edit it using SG's panel/file browser ... select config.php and click edit button. Am assuming they have one! 
2. has to do with debugging ... and what it says is true. But to see what issues there are for the install, debugging is the only way to actually know (better than my guesses). Once solved you can turn off debugging.
3. is cron job ... have to use whatever you have in SG's panel to setup a cron job for:
/path/to/php-cli/ /path/to/moodlecode/admin/cli/cron.php to run once every minute.
https://docs.moodle.org/311/en/Cron
If SG's one click installer didn't follow Moodle Doc's, suggest taking it up with SG support and ask why! :|
BTW, I host a moodle on GCE (Google) use Moodle's official installing docs just fine ... no issues.
'SoS', Ken