When I turn the key, I hear a "whirring" sound but the starter doesn't engage. Should I replace the starter or look elsewhere first? The battery is new.

 

That would be my guess.

 

Sounds like the starter solenoid is firing, the armature is turning but the shift lever is not moving. It may be able to be repaired but best to replace it.

 

Have you tried rotating your tires?

 

The starter drive could also have failed. It's function is to lock up in one direction so it turns the flywheel and then once started allows the flywheel to spin the starter drive while disconnecting from the starter drive shaft itself.
In Boxpin's diagram it's the over-running clutch.

But in either case, replacing the starter should cure your problem if we're understanding your noise correctly.

 

Take the starter off; look at the flywheel teeth. If they're all ground off and knarled, you need a flywheel. If the flywheel teeth look OK, renew (replace w/new or rebuilt) the starter.