There is a wide swath of providers. You could keep it easy to start with something like Linode or Digital Ocean, or you could go with Google Cloud or AWS.
Is your game server going to need to do anything besides present the game? If so I would consider a second server for the game server logic. It'll also help with partitioning resources. Properly setup, your website probably won't need much more than the specs of a Linode 1GB server that goes for $5 USD. A game server may need more.
GameDev.net's main servers are with LiquidWeb, but we'll be moving away from them in the coming months in favor of Google Cloud. We considered and evaluated AWS, but GCloud just makes more sense as a service for us plus our cost estimates led to it being less expensive. We also have a Linode VPS and a Digital Ocean VPS, so we're familiar with all of these. DO is "easy", but I have noticed network performance isn't as good as the others. We don't use Linode for anything strenuous so I can't comment on their performance.
Having said that, there are smaller cloud providers that might be a better option to keep costs low. You might want to consider those as a startup phase, and then when you're ready to scale up go with one of the larger providers. We actually advertise for several here on GameDev.net, so keep your eye out for those banners.