Writing an essay are we?

I can only answer the first part of your question with confidence because I've only studied in focus the Delian League - the reason why the Peloponnesian League formed in the first place - and Thucydides fairly thoroughly.

The answer to your question lies in Thucydides 1.67-88. It's a long set of passages and you should read it in detail, but the bare bones of it is that the Corinthians begged Sparta to help curb Athenian hegemony in Greece because they were under the impression that the Delian League was an increasingly imperialist organisation. And the Corinthians did have a point - Athens was increasingly throwing its weight around and denying rights to the member nations in favour of consolidating their own power. Halfway through the war they would annihilate every inhabitant of the island of Melos for not paying tribute. (This is called the Melian Dialogue and it's a very important piece of history about how the Athenians enforced their demands and what the Delian League had become - almost a vassaldom of the lesser states under threat of annihilation with a justification of might makes right, discussed explicitly in the Dialogue.)

Sparta, being a neutral party in the colonial conflicts between Athens and Corinth (this is pretty complicated and I won't explain it here because I'd be here all day) and being powerful as it was was the obvious choice for the Corinthians asking for help. In Thuc. 1.67-88 there is (invented but politically relevant and correct) dialogue between the three states. Sparta decides to side with Corinth and all states who have grievances with Athens flock to the cause, the League forms, and the war kicks off. Thucydides describes it as a containment war of sorts.

As for the administration, Sparta called the strategic shots according to Thucydides, but there's no detail about the nitty gritty. As for economic and geographic and religious stuff you'd need to look elsewhere. As far as I know there was no particular religious angle to either League. Some states held one god dearer than others (Apollo & Artemis for the Spartans and Athena for the Athenians) but IIRC there wasn't really a spilt or conflict on religious grounds at all. None that Thucydides mentions anyway.

I can't really point you to any books per se, but the first chapter of Thucydides precisely details how the war began and I don't think you can get much better than from the horses' mouth itself.

If you need a copy of the book Perseus is an online project that lets you find any line from any classical work you can name - but it uses translations from the turn of the century and it's clunky to use - but it's nice in a pinch. You can get it from bookstores as a Penguin for 10$, and PDFs are readily available. I personally use the Hammond translation (printed by Oxford) because I think it's the most readable (and Hammond is a fantastic historian, rest his soul.)