Learn Hardcore Icelandic: Mér líkar kaffi, en henni líkar te.

In Mér líkar kaffi, the subject is kaffi (nominative).
In henni líkar te, the subject is te (nominative). The verb agrees with that subject, not with the person in the dative.

Why is the verb líkar singular here? When do I use líkar vs líka?

Use líkar when the liked thing (the subject) is singular: Mér líkar kaffi.
Use líka when the liked thing is plural (or a coordinated list): Mér líka bananar; Mér líka kaffi og te. So number agreement is with the thing liked, not with the person.

How do I ask a yes/no question like Do you like coffee?

Use inversion with the verb first and the dative pronoun for the person:

How do I negate it?

Place ekki after the verb:

When do I use líka við instead of just líka?

Use líka við especially for liking people (being fond of someone):

Mér líkar við hana. (I like her.) Structure:
dative experiencer + líkar/líka
- við
  - accusative person. You can also use líka við for animals or, stylistically, for things, but with people it’s the standard choice.

Can I say Mér líkar hún to mean I like her?

No. Don’t use nominative hún there. Say:

Mér líkar við hana. (preferred for people) For things or activities, use bare líka: Mér líkar þetta lag.

Can I change the word order, like Kaffi líkar mér?

Yes, Icelandic word order is flexible for emphasis or topic:

Default and most common: Mér líkar kaffi.
Emphatic/topic-fronting: Kaffi líkar mér (vel). Both are grammatical; the first is more neutral.

Why is there no article on kaffi and te? What would the definite form mean?

No article (indefinite) expresses liking something in general:

Mér líkar kaffi = I like coffee (as a thing, in general). Definite form refers to a specific item or batch:
Mér líkar kaffið = I like the coffee (this particular coffee).
Henni líkar teið = She likes the tea (the specific tea).

How do I say I like doing something, e.g., I like to drink coffee?

Use an að-clause or a noun phrase:

Mér líkar að drekka kaffi.
Mér líkar að lesa. You can also say:
Mér finnst gott að drekka kaffi. (I enjoy drinking coffee; literally it seems good to me to drink coffee.)

What are the past tense forms?

Singular subject (3rd sg): líkaði
- Mér líkaði kaffi. (I liked coffee.)
Plural subject (3rd pl): líkuðu
- Mér líkuðu þessir drykkir. (I liked these drinks.)

Is líka also the word for also/too? How do I tell them apart?

Yes, líka can be an adverb meaning also/too. Distinguish by role and position:

Verb (liking): Mér líkar te.
Adverb (also): Ég vil líka te. or Mér líkar líka te. When it’s the main verb, you’ll see agreement (líkar/líka) and a dative experiencer.

How do I replace the pronouns with names?

Put the name in the dative:

Guðrúnu líkar te. (Guðrún likes tea.)
Jóni líkar kaffi. (Jón likes coffee.) Many feminine names take -u in the dative; many masculine names take -i.

What’s an alternative way to say I like coffee?

A very common pattern is to describe it as good to you:

Mér finnst kaffi gott. (Literally: Coffee seems good to me.) This is natural and widely used to express likes and preferences.

Which dative pronouns do I need after líka?

Can I leave out the experiencer (the person), as in just Líkar te?

No; you need to indicate who likes it. If you want a general statement, supply a generic experiencer:

How do I say I like coffee more than tea?

Use betur (better/more) and en (than):