Free hosting is always crap.
Don't use free forum hosting either because if you want to move, it may be hard to copy over the posts. And they are also crap.
You can use cheap shared web hosting but I prefer to use something like digitalocean.com or linode.com. Their cheapest plan is a good start. You will have to learn managing servers for those two though. If that's too much, find a good shared hosting plan.
We use about 100 GB of storage on the server, not too much actually. About 3-4 TB of bandwidth last time I checked.
Hive is a virtual machine with 8 GB of RAM, a quad core CPU and 500 GB of storage.
Use a free forum system to begin with, phpBB or SMF. Actually, if you dare to set up your own server or find some way to host it, I think Discourse (Google it) should be extremely good. It's free, you just need to host it somewhere (not shared hosting, most of the time because it's not PHP).