1. Forget about your idea, the maintenance of
the forum requires more abilities, than you have.
2. Take the phpBB software instead, which
is good enough, and free of charge (gratis).
3. You should ask not for the cheap hosting,
but for the reliable management service.
4. Read the ads on this forum.
Advertising Forums
Web Hosting Offers
Shared Hosting Offers
https://www.webhostingtalk.com/forumdisplay.php?f=4
Many people offer the managed hosting
service, and have good recommendations
from their happy customers.