Yes, both sentences are technically correct, if a little stilted. The 'It being sunny' phrase is correct but rather formal, and you need a future form to refer to the weather tomorrow. A more natural way of saying this is, for example,
'As it's going to be sunny tomorrow, we're going fishing.'
I'd use the 'going to' future in the first clause, because there is evidence (from the meteorologists' point of view) that this is is going to happen.
In the second clause, you could say 'going to go fishing', but it is a little wordy, so it's more natural to just say 'going xxxing', using a present continuous for a planned future activity.
The 'will' future tends not to be used for plans. We use it mainly for
- conditionals: 'If the it's sunny tomorrow, we'll go fishing.'
- decisions and offers made at that moment:
A: 'I want to try out my new fishing rod' B: 'OK, then. We'll go fishing tomorrow.'