What is wrong with my bike's gear shifting?

[Marlogism]

my brand new mountain bike's shifting is very bad. the chain keeps moving between gears by its own and it doesnt shift right when i want it to. i must keep holding on the shift down hand a bit so that the chain stay in place.
i thought messing a bit with the limit screws would solve it but i think i have over done it and made it worse.
what should i do?