I register through Hover.com (owned by Tucows, who is actively fighting against SOPA). Enter code: HONEST and save 10% :P. They may seem a little bit more expensive, but all their domains include domain privacy...so if that is important to you, they are probably the most affordable around. Their site is nice and clean, easy to navigate, and not filled with a bunch of advertisements :).
Which host you choose can be a bit trickier as different ones allow different things (some won't allow Arcade type sites to be set up for instance). I use SquareSpace, but there are limitations...one of which is you cannot host Forums on the site, or Host any file larger than 20MB. But as zutri said, free hosting isn't very good, and shared hosting isn't much better...usually max user connections is between 15-25), not to mention the slower load times from so many sites shared on the same servers.
If you can't quite afford the a $20 per month site, the best shared hosting I was able to find is from a small independant host MyBBWebHost.com (named after the free Bulletin Board MyBB). They don't overcrowd their servers, and they allow up to 200 max user conenctions. For this reason I decided to host the Fun Chowder Community there. And they don't promise unlimited web storage and bandwidth, which in my opinion is much better, since I would rather have 20GB web space and 200GB transfer that works well than unlimited that doesn't.
Not 100% sure about hosting an Arcade, can always send them an e-mail. They respond quickly for a small host, but my guess is that it would be allowed. Once you start getting enough traffic to strain this host, then you can think about getting dedicated or cloud hosting.
Hover.com for domain registration (if domain privacy is important to you)
MyBBWebHost.com if you can only afford shared hosting. They don't have a site builder per say, but they do have softaculous, which can auto install over 250 scripts, such as Drupal or Wordpress.
edit: With http://www.webwiz.co.uk, as Tom mentioned earlier is the host for Scirra, at least you know you can set up an Arcade, and you can upgrade your package as you grow. Not sure what the max user connection limit is for the shared hosting though.