If your needs are fairly low on the processing side, you can snag a cloud VPS on LowEndBox for five or six dollars a month. Quality is highly variable ofc, but I’m reasonably my happy with mine.
No AWS, etc (though I don’t know offhand where the actual box lives), SSH access defaults to a key, and the rest (firewall, reverse proxy if you like, and all the other best practices) are but an apt-get away and a quick searxng to find and dissect working configs.
Incidentally, searxng is a good place to start- dead easy to get rolling,and a big step towards degoogling your life. Stand it up, throw a pretty standard config at nginx, and do a certbot —nginx -d search.mydomain.com - that all there is to it.
YMMV with more complex apps,but there is plenty of help to be had.
Oh…. Decide early on if anonymity is a goal,or you’re ok tying real life identity to your server if someone cares to look. Register domains and make public facing choices accordingly.
Either choice is acceptable if it’s the right one for you, but it’s hard to change once you pick a path.
I’m a big fan of not hosting on prem simply because it’s one more set of cables to trip over, etc. But for a latte a month in hosting costs, it’s worth it to me.