I've made a list of a lot of services and such
here to try and help out with people like you starting out.
If you want something cheap, I would say that Namecheap Shared Hosting is fairly good for things like faucets, but I have been told that it sometimes can be somewhat slow loading. It isn't cripplingly slow by any means, but if you want lightning fast speeds it probably isn't best. If you want absolute control and speeds I would say get a VPS, but that seems a bit OTT for something like a Faucet.
I would recommend using FaucetBox as there is significantly more traffic to the sites on their list and the script is easier to edit if you want to. However, a payment provider such as Paytoshi works just as well.
I would try to get a custom theme for your faucet if possible. It attracts more users to your site if it is easy to use, can open more opportunities to revenue and can also help to stop bots. As for stopping bots, I would recommend reading through the last few pages of the
FaucetBOX.com Discussion thread, as methods of using Captchas and such to stop bots are discussed.
Thinking about using faucetbox.