High revvs when changing gear

I usually drive a diesel but have recently bought a petrol car and find it to have high revvs when changing from low to high gears but not when changing down gears. When I press the clutch in to change and take my foot off the gas it sounds as though I've still got my foot on the gas and it seems very quick to pull away too, should I get it looked at and what is it likely to be / cost?