The verb guardar means to put a thing in its place so as not to damage it, not to lose it and to keep it safe. In other words, to put things in their place to keep them safe and protected:
Tengo que guardar la ropa = I have to put the clothes away (I'm going to put the clothes in the closet, so everything is neat and in its place)
Por favor, guarda las llaves del coche = Please, keep the car keys (Please, put the car keys in a safe place because we don't want to lose them)
Meter, on the contrary, means putting one thing inside another, in a closed place:
¿Por qué siempre metes las manos en el bolsillo? = Why do you always put your hands in your pocket?
Por favor, mete las llaves del coche en mi bolso = Please, put the car keys in my bag.
In many contexts, guardar means meter en su sitio (put smth in its place):
¿Puedes guardar la comida? = ¿Puedes meter la comida en el frigorífico? = Can you put the food in the fridge?
Después de jugar, tienes que guardar los juguetes = Después de jugar, tienes que meter los juguetes en su sitio. = After playing, you have to put the toys in their place.
However, unlike guardar, the verb meter requires an adjunct of place, i.e., where you have to put the object:
CORRECT: Guarda bien las llaves
INCORRECT: *Mete bien las llaves
The only exception is in football, where meter gol means to score a goal. For example:
Messi ha metido un gol al final del partido
Messi has scored a goal at the end of the game
The verb meterse means dedicate yourself, start participating:
Voy a meterme en un proyecto = I'm going to get into a project (I am going to start participating in a project)
The expressions meterse en problemas / meterse en líos mean start having problems:
Juan siempre se mete en líos = Juan always gets in trouble; Juan is always looking for problems
Finally, learn these expressions with the verb guardar:
guardar un secreto = keep a secret (not to tell a secret to anyone)
guardar silencio = keep silence
guardar las distancias = keep a distance (stay away from a person)