Disclaimer - I practise mindfulness but I'm not an expert.

Yes it's normal to feel agitated initially. If life is focussed on doing and achieving stuff then deliberately stopping and deliberately not doing anything feels wrong. But isn't it interesting that your mind reacts that way?

I think of mindfulness as a way to be able to observe my mind, to notice what it does, to notice what feelings I have (like agitation sometimes).

What specific mindfulness exercise are you doing when you get agitated? Focussing on the breath can be helpful because it tends to calm people.

I think it is completely normal, I have been meditating daily for 5 years and my mind still gets agitated and restless during meditation, but overall, day to day, it has become lots easier to focus and the day to day agitation and restlessness has been massively calmed.