First piece of advice: Relax, don't sweat it too much. I 'hosted' my first side project on a Raspberry PI. My second was on an AWS EC2 instance on the tiny free tier. Nothing wrong with having the db and the app on the same server (in my case Python and Postgres). The projects were mostly for fun though, and I was not worried about scaling or anything like that, but it was quick, cheap and easy.
If you are worried about scaling (which would be a nice problem to have), you can still start out on AWS (or another provider's) free tier, and then just upgrade the instance if you start getting an uptick in use. If you are lucky enough to out grow vertical scaling, then you will have plenty of motivation to figure out the horizontal part. Just carve out an afternoon to sort it out. And if you are really ambitious, try setting your app up on multiple providers. After the first one is up, figuring out the others should be easier (cause they all basically do the same thing), and then you can play around with them and stick with the one you like best. Might cost a little more initially, but could help put your mind at ease.
Regarding security, just make sure you are salt+hashing users passwords. You can worry about the other stuff later. And as others have mentioned, if you do one thing, make sure you backup your database. You can start by just doing it with a daily crontab (or even do it manually if you really need to). It your app crashes, you can always restart it, but if you loose the db, you are hosed.
Also, you might want to consider getting some test users (friends, family) to poke at it before releasing it into the wild. This will help you surface any bugs. If this is your first web app, I would rate your odds of having issues large enough to ruin your launch as: 95% chance due to bugs | 4.5% chance due to scaling issues. | 0.5% chance due to security issues. If a site is getting the HN hug of death, it usually peaks my interest more, and I will bookmark and come back to it. But if its buggy as hell, I loose interest right away.