I am trying to find a decent espresso machine for my mother's birthday, but I am a kind of confused with the variety of choice and advices about these machines. Actually I do not know what is easier to use - an espresso machine or something like stove top coffee maker?

A fully automatic espresso machine will be much easier to use than stove top maker But iIf you practise a lot with stove top, you will be able to make the same quality espresso like in espresso machine.

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I am making the very first steps in coffee world

Yes, I purchased fully automatic machine and my mother was happy to receive it. She is now playing with her new Saeco Vienna digital

i have both and love them as much as each other.