Cutting out bottom plate for new door in existing partition
 

I'm installing a standard inside door along an existing interior wall. The wall is bare with 2x4's sixteen on center. I've removed one 2x4, leaving the beginnings of the new door frame.

What is the best way to saw the bottom plate without cutting through the subfloor? My circular saw would leave some board sticking out on either side.. is this a job for a hand saw or is there a better way?

Hand saw or reciprocating saw.

Is this a bearing wall?

Sounds like you do not yet own a sawzall (reciprocating saw).
This job would be a great excuse for you to buy one!
You will find many uses for it in the future.

As Joe mentioned above, make sure that you use proper framing techniques
to support the load above.