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So... I need to cut a bigger hole in this wood.

Which tool? Someone suggested a small circular saw

NO! That is the wrong tool for sure.

and since I have
zero tools suggested I buy a cordless combo kit from Sears (Craftsman).

OK... I look at the tool kit and see a Reciprocating saw!?
Would that not be a better choice to use?

Better choice, but not the best tool.

Three options. Small hand saw like a keyhole saw, a Rotozip with a cutting
bit, or a jig saw.

If you decide to go with a Rotozip, practice on a scrap first as you can
easily get carried away and do damage. Jigsaw is easier to control and will
be a more useful tool to own in the long run.