The comparative
The comparative is formed in English by adding "-er" to an adjective or adverb, or by preceding it with "more", according to the length of the adjective - i.e. "bigger", "more difficult", "more easily".
Unfortunately some of the most commonly used German adjectives and adverbs have comparatives which are exceptions to this rule: either the vowel "modifies" - i.e. adds an umlaut - or a different word is used entirely:
The comparative using "mögen"
As the optional adverb used with "mögen" is "gern", you need to use "lieber" (= the comparative of "gern") if you wish to use this verb to say that you prefer something:
Grammar 15: Comparative with "mögen"
Wir mögen Berlin (gern).
("We like Berlin.")
Wir mögen Berlin lieber.
Wir mögen lieber Berlin.
("We prefer Berlin.")
Welchen Wein mögen Sie lieber?
("Which wine do you prefer?")
Ich mag Rotwein lieber.
Ich mag lieber Rotwein.
("I prefer red wine.")
By extension, as you use "gern(e)" to describe what you like doing, you will also use "lieber" to describe what you prefer doing:
Grammar 16: Comparative with verbs
Ich höre Händel gern.
("I like listening to Händel.")
Ich höre Händel lieber.
Ich höre lieber Händel.
("I prefer listening to Händel.")
Welchen Wein trinken Sie lieber?
("Which wine do you prefer to drink?")
Ich trinke Rotwein lieber.
Ich trinke lieber Rotwein.
("I prefer to drink red wine.")
The comparative using "gefallen"
As the optional adverb used with "gefallen" is "gut", you need to use "besser" (= the comparative of "gut") if you wish to use this verb to say that you prefer something:
Grammar 17: Comparative with "gefallen"
Diese Stadt gefällt uns (gut).
("We like this city.")
Diese Stadt gefällt uns besser.
("We prefer this city.")
Was gefällt dir besser?
("What do you like better?")
Das Armband gefällt mir besser.
("I prefer the bracelet.")