Second Goudrons - badly worn clutch.
Delicate question to ask your sister, but did she 'ride' the clutch?
You could just try 'pumping' the clutch two or three times in quick succession before selecting a gear (from stationary). If you find that the lever slips smoothly in after doing this, it would suggest a hydraulics problem - but from what you've described so far it looks like excessive clutch wear. Maybe gentle linkage adjustment needed also.
Who should I take it to to get it looked at? Fiat?
Typical mechanics would see me, young and clueless and tell me it needs done whether it does or not. To be fair, Fiat employees gain nothing as they get paid a set rate surely? So I'd trust their opinion more perhaps.
Should I still go ahead and pay to get the timing belt done with this possible issue or prioritise the clutch?