If you floor it from a standing start an auto should rev to perhaps 2000 / 2200 RPM and the revs should then increase following the car's speed until almost the red line where it will change up.
It might feel a little more sluggish than a manual, firstly because 1st gear in an automatic gearbox is relatively high geared (equivalent to 2nd in a manual usually, however, the torque converter does add a slight variability to bring the ratio back down) and secondly because the engine has to pick up from idle speed, under load, until torque is transferred to the wheels. In a manual, the driver can "anticipate" the start and hold the engine RPM above idle without load, and can also transfer torque to the wheels much more shaprly using the clutch, which certainly feels faster.
I would have concerns if the engine revs at the instant you floor it rise to more than around 2K rpm, or fail to reach this valve, waver around as the car pulls away, or falter. In addition, when the box shifts up a gear the revs should drop smartly. If they linger before dropping, or the box feels like it's "slipping" I'd do a fluid and filter change ASAP (in fact, I'd do one anyway if you don't know when it was last done).
Autos do respond differently to manuals, and will feel odd to drive if you're not used to them. In a manual, engine RPM depends only on the gear you are in and the speed you are doing.
At low speeds, and automatic gearbox has a constantly changing gear ratio so engine revs will also depend on engine load (how much throttle you're using). This makes it feel like you're driving a manual with a slipping clutch at first.
Kevin