Sirius is very smart. The way I understand it is this:
Everyone (including Dumbledore) thought Sirius was the secret keeper.
Everyone (including Sirius) knew that the Secret-keeper betrayed the Potters.
Sirius and the Potters were the only ones who knew Pettigrew was the one who betrayed the Potters and after their death Sirius went looking for Peter.
When Sirius finds Peter, Peter yells for the whole street to hear that Sirius betrayed Lily and James. Then, before Sirius could curse him, he blows apart the street with the wand behind his back, killed everyone within twenty feet of himself — and sped down into the sewer.
Sirius thinks Peter killed himself because his master died. Peter would obviously not admit to anything since his master is gone.
- Knowing what awaited him in Azkaban surely he would have at least protested his innocence rather than go to prison?
No one would believe him, he has no evidence and all available evidence points at him. He understands this and he deems protesting as a waste of time and energy.
- And then why wait all that time before slipping out as a dog if he could've done it sooner?
He has no reason to get out. Everyone thinks he's guilty. He would be a fugitive. He got out only when he understood Harry was in danger.
- Yes, he does say that the newspaper article about Wormtail on holiday with the Weasleys triggered his something inside him, but if he was innocent would this not have been a logical solution in the beginning?
The newspaper doesn't trigger something in him. He sees the rat on Ron's shoulder, notices the missing finger, puts it all together and figures out Wormtail is still alive and going to Hogwarts!