Both “like more” and “like better” (as in the sentence “I like apples more/better than oranges”) are widespread, but “like more” is usually considered more formal and “like better” more informal (some British English speakers incorrectly consider “like better” to be an Americanism, even though “like something better” predates “like something more” by several centuries and is common in British literature). To summarize:
If you use either of the two, you will be understood. If you stick to “like more”, you also don’t run the risk of sounding too informal (or colloquial in the UK).
Expressions “to like most” and “to like best” seem to be acceptable both in American and British English. Some speakers use the variant with “the”, i.e. “to like the most/best”, but others consider it less grammatical. It is therefore advisable to stick to the variant without “the” (which is also much more common in literature):