Hi Smelval!
'Gaze' normally has a positive meaning but can also sometimes be neutral.
"She gazed at the night sky, full of wonder."
"He gazed into his girfriend's eyes, wondering what she was thinking."
'Stare' can be neutral but is often negative.
"He stared at the man, making him uncomfortable."
"She stared at the bill in disbelief."
They both suggest that you look at something for a long time without looking away, but generally tell us if it's in a good way or a bad way!
I hope that helps!
Mark