i have always been a customer of godaddy.com and i stand by their quality of service and good prices. Im pretty sure there's companies that do it for cheaper but i personally like godaddy because they have a very strong platform of software and support.
I would start off by looking at what forum engine you are going to use, if you want a free one, phpBB is a great open source forum engine and most hosting websites like godaddy will install it for you, all you gotta worry about is getting familiar with phpbb and getting a nice template to suite your looks.
vBulletin is one of the best but also very expensive as in like $200 and up for one license but it has tons of nice features and really easy to setup and get nice looking templates.
I would recommend phpbb if you dont wanna pay
phpBB • Free and Open Source Forum Software
if you are interested in Vb
https://www.vbulletin.com/ (the forum you are browsing right now is run by vB)
They once you decide that you can go out and by your domain from a website (you can google it, my recommend is godaddy) and then once you have your domain you have to purchase hosting. Theres so many websites out there i can't list them all, but im pretty sure a quick google search will get you there. (like i said godaddy, check it out)
I was just runing a forum not to long ago and i used phpBB, and i had fun with it and learned allot, i can always point you in the right direction if you need.
oh and when you get hosting, get the basic plan because you wont expect allot of traffic hence you wont need allot of space. 10gb a month will do you fine, and when you get bigger you can always upgrade.
Good luck