Just tried installed Statamic on a DO droplet. I have no previous meaningful experience with Laravel. A few notes and questions.
What are the permissions supposed to be?
For Laravel only storage and bootstrap/cache need web-server (www-data) write permissions, but in my quick experiment it seems that all of Statamic needs to be be writable by www-data. Since loose permissions is not a good thing, is there some guide you have?
Where is the email setup? I am vary keen on buying this, but need to test user permissions first, and have not been able to get a second user created.
Could I suggest that you give the apt packages for the server requirements on the requirements page. Ubuntu is no doubt the primary platform it is installed on.
sudo apt install php-common php-bcmath php-json php-mbstring php-tokenizer php-xml php-gd
Also, this markdown editor may have an issue - I could not get a list to work in the preview.