Starting from idle, acceleration lowers RPMs

Starting from idle, the rpms decrease to almost or complete engine stall when pressing the gas to accelerate. If I floor it, acceleration responds and gear change delays way beyond normal rpm range, up to almost 6000 rpms. Less severe gear change delays occur after the first gear shift. Those who know or fixed similar issues: what are the MOST likely causes to inspect for diagnosis first? Note: I may buy a diagnostic tool if I think it will be worth it.