Do I have to collect each bit of belief manually?

Solution 1:

Yes. There is no way to automate the belief collection; no matter what, you have to keep clicking.

Just like it's predecessor, Populous, Godus mandates clicking, and lots of it.

So click on, brave god! Click forever!

Solution 2:

Initially, yes. You must click. You must click a lot in order to collect belief. You can also hear a groovy Mozart concerto in the process. This applies to the primitive age of development.

Later, once you've developed some, when you can put down a statue and create a settlement, belief will gather at that statue. Then you can click a bit less often to collect a lot of belief at once. It will still involve clicking and a lot of it, but with greater yields than before.