Learn Hardcore Dutch: Wat ik het liefst doe, is ’s avonds rustig lezen. - What I like doing most is reading quietly in the evening.

Because this is a free relative clause (no explicit noun antecedent) and it follows a superlative (het liefst). In Standard Dutch, after superlatives and after words like alles/iets/niets/veel/weinig, you use wat, not dat.

Correct: Het liefste wat ik doe, is …
Also correct (with an explicit noun): Het boek dat ik lees … Note: Many natives say het liefste dat ik doe in speech, but in careful writing wat is preferred here. A very formal alternative is hetgeen: Hetgeen ik het liefst doe, is …

Why is doe at the end of Wat ik het liefst doe?

Because that clause is subordinate. In subordinate clauses in Dutch, the finite verb goes to the end (verb-final). So: Wat (introducer) + ik (subject) + het liefst (adverb) + doe (finite verb at the end).

Why is the main verb is in second position?

Dutch main clauses are V2: the finite verb comes in second position. The entire clause Wat ik het liefst doe counts as the first constituent, so the main verb is must come next:

[1] Wat ik het liefst doe, [2] is …

Do I need the comma before is?

It’s optional but recommended for readability. You can write either:

Wat ik het liefst doe, is ’s avonds rustig lezen.
Wat ik het liefst doe is ’s avonds rustig lezen.

What is het liefst grammatically, and why not het liefste?

Het liefst is the superlative adverb of graag: graag → liever → het liefst. It modifies a verb (what you prefer to do), so it’s adverbial and takes no -e.
Het liefste (with -e) is an adjective used before a noun: mijn liefste tante.

Could I say Ik lees het liefst ’s avonds rustig instead?

Yes. That’s a very natural, slightly simpler word order. Other good variants:

Het liefst lees ik ’s avonds rustig.
Ik lees ’s avonds het liefst rustig. They all mean the same; the differences are mostly about emphasis and flow.

Why is there no te before lezen?

Here lezen is a bare infinitive used as a predicate complement after is (What I most like doing = reading). Dutch does not use te in this construction. Adding te would change the meaning (e.g., makkelijk te lezen = easy to read) or be unidiomatic here.

What exactly is ’s avonds, and how is it different from in de avond?

’s avonds is a fixed expression meaning “in the evening(s),” from a contraction of archaic des avonds (genitive). It’s very common and neutral.
In de avond is possible but tends to sound more specific or formal; for habitual time-of-day, Dutch strongly prefers ’s ochtends / ’s middags / ’s avonds / ’s nachts.

Where can I put rustig? Is rustig lezen the only option?

With an infinitive, the adverb normally comes before it: rustig lezen. In a finite main clause you’ll typically have Time–Manner order:

Ik lees ’s avonds rustig. You can move rustig for emphasis, but lezen rustig (with the infinitive after the adverb) is the default in the infinitival phrase.

Why is it rustig and not rustige?

Rustig is used adverbially here (modifying the verb lezen). Dutch often uses the base adjective form as an adverb. Rustige (with -e) is the attributive adjective before a noun: een rustige avond.

Is doen really necessary here? Could I say Wat ik het liefst is …?

You need a finite verb inside the relative clause. Wat ik het liefst … without doe is incomplete. Doen works as a support verb to express “the activity I most like doing.”
Note that Wat ik het liefst lees would mean “What I most like to read,” which is a different meaning.

How do you pronounce the tricky parts like ’s avonds, het liefst, and rustig?

Approximate Dutch pronunciation (Netherlands):

’s avonds: [s ˈaːvɔnts]
het liefst: [ət lifst] (the vowel in het often reduces)
rustig: [ˈrʏstɪx] (final g as ; in Flanders often [ɣ]) The ’s is pronounced as just an [s] attached to what follows.

Can I say Het liefste wat ik doe, is … or use het allerliefst?

Het liefste wat ik doe, is … is fine; it treats het liefste as a noun phrase. After this superlative used substantively, wat is the standard relative.
het allerliefst intensifies the superlative: Het allerliefst lees ik ’s avonds.

Is there any nuance difference between Wat ik het liefst doe, is … and Ik lees het liefst …?

Slightly. Wat ik het liefst doe, is … sounds a bit more deliberate or rhetorical (common in explanations or essays). Ik lees het liefst … is more direct and conversational. Both are perfectly natural.