C-West Civic

Hey guys, 03 Accord starting issues. Started months ago with the occasional slow cranking engine. Yesterday went to start and it wouldn't crank over to start car. The starter engaged, and got a few turns of the engine but then nothing. Had the battery tested last night and was said that it was slow at taking a charge, so I bought a new battery. Put it in this morning and same thing. I ruled out battery ( since new ) starter ( since it's engaging ) alternator ( since I don't think it's needed to actually start car ).

Any ideas as to what I should at look at? Thanks