Nikko is famous for its hot springs. I was planning to visit it with the intention of enjoying some good onsens.
Are there different kind of onsens available there i.e for instance some are there in a ryokan or some are free for the public etc?
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The town of Nikko itself, where the Toshogu shrine is located, is not in a volcanic area and thus does not have any onsen hot springs. You have three choices in the vicinity:
All three are major tourist areas and have accommodation running the gamut from Western-style hotels to fancy ryokan to simple B&B/minshukus. I don't believe there are any free baths, but many ryokan/hotels open their door to visitors during the day for a small fee (around ¥500 is common).