The shifter mechanism is one of the easier bits to test. Click it up and down through its range a few times, and make sure it hits all the stops with no excess friction.
If the shifter misses a click, it means you either have the wrong shifter on the bike (e.g., an 8-speed shifter on a 9-speed setup), or the shifter has a broken internal tooth. Either issue is resolved by replacing the shifter.
If there is excess friction, you can try to spray PB Catalyst into the shifter itself to ease any thickened grease in the mechanism. If this does not help, you can replace the shifter cables and housing.
Part and Labor Estimates:
New Shifter: $30-100 and up per shifter.
New Cables and Housing: $25-$40 per shifter
Labor for new cables/housing or install of shifter: $25 and up per side.