My German isn't very good, but from what I understand 'gern' is an adverb while 'mögen' is a verb. Mögen is conjugates irregularly to 'mag'. Ex. Ich mag etwas. = I like something. Ich mache gern etwas. = I like doing something.

mag is used to say that you like something or someone. for example... I like him. Ich mag ihn. Ich mag Brot. Ich mag Kaffee. Also note it takes the accusative case. conjugate mag in the following way. ich mag, du magst , wir mögen, Sie mögen, ihr mögt, er/sie mag there is also past tense, hat gemocht and simple past mochte, ich mochte Kaffee. I liked coffee. Ich mag Kaffee. gern is used for when you like doing something. Ich spiele gern Tennis. Ich koche gern. note that gern comes after the verb* or if you want to say you don't mind doing something... aber gern! or just gern. und so weiter. gern sometimes appears as gerne, which I don't think changes anything. I hope this helps and maybe any native speakers can clarify anything that I have missed.