You could say to the supplying dealer that the gearchange is not right, describe the symptoms as you have done here, and ask them to investigate. Don't be fobbed off with "they all do that," because they don't.

In reality, the Sale of Goods Act states that any fault which develops within 6 months of you purchasing the car is assumed to have been present when the car was sold to you - so you do have extra comeback.

It does sound like the gearbox actuators need a fluid change, as a minimum.

Hi Joanne

Welcome to the forum

An "Easytronic" is essentially a manual gearbox and clutch assembly with some electronic "gubbins" ( motors / relays and things like that) that do the task a human would do in a manual car ie: pressing the clutch, choosing a gear, moving the gearbox shafts to select the gear and then letting the clutch re-engage the engine power while balancing the throttle input.
Manufacturers tend to use automated manuals on lower power / city cars to maintain fuel efficiency and not sap to much power from the low power engines.

The decision to change gear is done by electronic control units (computers etc) that measures things like engine speed / are you braking / throttle position / engine temperature etc to decide what gear it thinks you want and suits the current conditions (ie: it won't let itself change from 4th to 1st as this could damage the engine)

The Vauxhall Easytronic has a mixed reliability history and has anecdotally on various forums more than its fair share of failures.

Automated manuals can have a bit of a "jerky" change and be a little indecisive when they change which can cause revving but it does sound like yours is a bit beyond "normal" jerkiness.

As you have an auto licence only (becoming more common due to hybrid and electric vehicles) , I'll explain kangarooing when pulling away in a manual car is caused by a mismatch of engine speed to clutch position so it does sound like either the clutch needs some work or the gubbins doing the changing need work to get it changing a little more smoothly.

If you get a good one of these they are every easy to drive and almost as smooth as a "proper" old style torque converter automatic.

As others have indicated, first action should be to alert the seller of the issue and give them an opportunity to fix it. If it is a reputable dealer they "should" be willing to listen.
As you indicate they said a 31 day warranty, this is contrary to the current sale of goods act which is law in the UK and it sounds like they may give you excuses rather than fixes. For a major fault like this, the SOGA act DOES apply, despite it being an older car.
Availing yourself of SOGA rules regarding used cars might help you have confidence before approaching seller

Good luck and please do come back and let the forum know what happens