Learn Hardcore Icelandic: Ég kenni henni að nota tölvuna.

Henni is dative singular of hún (she). The verb kenna selects the person being taught in the dative, so you say Ég kenni henni ... (literally: I teach to-her ...).

What does að do before nota? Is it like English "to"?

Here að is the infinitive marker introducing nota (use). It works much like English "to" before a verb. After verbs such as kenna or hjálpa, Icelandic uses að + infinitive to express teaching/helping someone to do something.

Why is it nota, not notar or notað?

After að, the verb is in the infinitive, which here is nota.

Notar is a present-tense form (e.g., Hún notar tölvuna).
Notað is the neuter past participle (e.g., Ég hef notað tölvuna).

Why is it tölvuna and not tölvu?

Tölvuna is the definite accusative singular of tölva.

Nota takes its object in the accusative.
Icelandic marks definiteness with a suffix: tölvu (acc. sg. indef.) + -na = tölvuna (the computer).
If you mean “a computer” (indefinite), say að nota tölvu.

What’s the basic word order here? Can I say Ég kenni að nota tölvuna henni?

Main-clause word order is verb-second: Subject + finite verb + objects/complements.
So: Ég (subject) kenni (finite verb) henni (indirect object) að nota tölvuna (infinitival clause).
Ég kenni að nota tölvuna henni is unnatural. You can front elements for emphasis, e.g., Henni kenni ég að nota tölvuna, but the pronoun normally precedes the að + verb phrase.

How does kenna conjugate?

It’s a regular weak verb:

Present: ég kenni, þú kennir, hann/hún kennir, við kennum, þið kennið, þau/þær/þeir kenna
Past: ég kenndi, þú kenndir, hann/hún kenndi, við kenndum, þið kennduð, þau/þær/þeir kenndu
Past participle: kennt
Progressive: Ég er að kenna; Perfect: Ég hef kennt

If I change the person, which pronoun forms do I use after kenna?

Use the dative forms:

me: mér — Kenndu mér ...
you (sg): þér — Ég kenni þér ...
him: honum — Ég kenni honum ...
her: henni — Ég kenni henni ...
it (neuter): því (for things)
us: okkur — Hann kennir okkur ...
you (pl): ykkur — Hann kennir ykkur ...
them: þeim — Hún kennir þeim ...

Can kenna also take a direct object like “teach someone something”?

Yes. Pattern: kenna e-m e-ð = teach someone something. The person is dative; the thing taught is normally accusative (or an að + infinitive clause). Example: Ég kenni honum lögin (I teach him the laws).

Could I also say Ég kenni henni á tölvu? How does that differ from að nota tölvuna?

Yes. Kenna e-m á [device] means “teach someone how to operate a type of device” (general ability).

Ég kenni henni á tölvu ≈ teach her how to use a computer (in general).
Ég kenni henni að nota tölvuna focuses on using the computer (and with the definite, a particular one).

Why not use til að (e.g., kenni henni til að nota)? I see til að a lot.

Til að introduces purpose (“in order to”): Ég fór til að kenna henni .... After kenna, you use plain að before the infinitive: Ég kenni henni að nota ..., not til að nota.

How do I negate the sentence?

Place ekki after the finite verb: Ég kenni henni ekki að nota tölvuna.

How do I say it in the progressive “I am teaching her ...”?

Use vera að + infinitive: Ég er að kenna henni að nota tölvuna. (Simple present Ég kenni ... can also cover ongoing actions.)

How do I say “Teach me to use the computer”?

Use the imperative: Kenndu mér að nota tölvuna. (plural/formal: Kennið mér ...)

If the computer is already known, what pronoun replaces tölvuna?

Use the feminine accusative pronoun hana (it = her, for feminine nouns): Ég kenni henni að nota hana.

Is this the same að that means “that” or the preposition “to/towards”?

Same spelling, different functions:

Infinitive marker: að nota
Complementizer “that”: Ég veit að hún kemur
Preposition “to/towards/at” (takes dative): fara að húsinu
Context tells which að it is.

Any quick pronunciation tips for this sentence?

Ég: roughly “yeh” with a soft voiced “gh” at the end.
ð in að: like the th in “this”.
ö in tölvu-: like the vowel in British “bird,” but with rounded lips.
Double consonants (nn in kenni, henni): hold the consonant a bit longer.
Stress is on the first syllable of words.

Can I drop að before the infinitive like English does after modals?

After modals (e.g., vilja, geta, mega) you do not use að: Ég vil nota tölvuna. But after kenna, you must use að: Ég kenni henni að nota tölvuna.

Why is it Ég at the start and not Mig?

Ég is nominative (subject case), required for the subject of kenna. Mig is accusative and would be used as a direct object with verbs that take the accusative (e.g., Hún sér mig). With kenna, the person taught is dative: Hún kennir mér (not mig).

What changes if I want plural “computers”?

Use the accusative plural:

Indefinite: tölvur — Ég kenni henni að nota tölvur.
Definite: tölvurnar — Ég kenni henni að nota tölvurnar.