Mmm, "gaze" and "stare" are two ways to describe "looking at". If I "gaze" at something or someone, I think of it as a look while thinking of something nice (or naughty). If I "stare", I will be looking at something or someone hard (maybe I wouldn't blink). It usually has my full attention (like I stare at a nice sports car as it zooms by, or looking hard at someone's very unusual anatomical feature). They are usually not interchangeable, as the connotations are different.
In your example, if my mum stared at me and told me something, I think it would usually be a bad thing. If she gazed at me, maybe I wouldn't be so nervous.
"Peek/ peep/ glance/ glimpse" are descriptions for a quick look at something, maybe from behind an object. In glimpse, it has an additional meaning of looking at something very briefly as it comes into view (maybe by accident- a glimpse of a shadow behind the curtain at night is always scary).