Is my chain acting normally ?

I see my chain "sticking" to the gear then bumping upon "releasing".
I hear my chain doing a weird noises.
I feel strong vibrations.

I ride bikes for decades now, but I'm not great at maintenance (that's why I ride this simple bike). I am too affraid to ride it… Am I tripping ?

Every pieces are brand new except from the frame. I think I got the correct 1/8 (?) standard for the chain and gear.
I also tested with another wheel and it's as weird for me.

I went to a local boutique and they told me the chain wasn't tight enough… Which… was true that day cause I forgot to set it correctly before going to their shop. I ran to my house ashamed and never went back.