I currently run a fansite powered by Wordpress with a SMF forum with almost 2 million posts. It's not as busy as it used to be though. We get about 4,000-5,000 visitors a day with 40K page views.
I'm using 12GB disk space plus 1GB for the mysql database. I'm mostly concerned with CPU usage. I was on a VPS with Inmotion Hosting 3 years ago and they were constantly complaining about me using too much resources so I had to leave. Someone hosted us on their own server but now I'm possibly looking for a VPS. I'm on a budget too. Can't really afford it and the ads we have on the site don't really generate that much money.
Did you tried to work with them to investigate what is the issue with the site that is causing more cpu usage than the other resources like ram etc?
misconfigured plugins or theme can cause the issues as well.
Not a friend, it's like a fansite network. There's probably 100 other sites on the same server as mine. They're cutting edge servers though, I imagine. Very rarely had any problems with them and they've never complained about us using too many resources. It's free hosting in exchange for a banner ad. Only reason I'm considering leaving is that they're now enforcing an ad bar across the top of the page which is something I'm totally against.
You can have a decent VPS node for arround $50.
You can even get a managed node for a little over $50 with enough CPU to run your forum, but with the agreement you are going to install some caching over it
Consider Memcached or Varnish or CloudFlare. That can have quite an effect.
I would recommend you for managed VPS server to get it monitored 24/7 with a good VPS configuration with enough ram and cpu to handle 4000 - 5000 visitors. Managed VPS mostly include to fight spam coming to your server and block them immediatly. $50 would good enough to have such plans.