Learn Hardcore Swedish: Efter vilan känner jag mig lugn.

You could say Efter vila (after rest), but that sounds more general or abstract, like “after (having) rest” as a concept, not a specific occasion of rest. In everyday speech, Efter vilan is more natural when you mean a particular rest or nap.

Can I also say Jag känner mig lugn efter vilan instead? Is the word order important?

Yes, Jag känner mig lugn efter vilan is also correct.

Swedish has the V2 rule: the finite verb (here känner) normally comes in second position in main clauses.

If you start with the subject:
Jag (1) känner (2) mig lugn efter vilan
If you start with a time expression (Efter vilan):
Efter vilan (1) känner (2) jag mig lugn

So both are fine:

Efter vilan känner jag mig lugn. (focus on after the rest)
Jag känner mig lugn efter vilan. (neutral focus on I feel calm)

The meaning is basically the same; the difference is just in emphasis and style.

Why do we say känner jag mig instead of just känner jag or är jag?

Two separate points here:

Why not just känner jag?
The verb känna (to feel) is reflexive when you talk about how you yourself feel in your body or mind. You must add a reflexive pronoun:
- Jag känner mig lugn. = I feel calm.
- Jag känner honom. = I know him. (here känna is not reflexive)
Without mig, känner jag lugn is ungrammatical.
Why not use är (to be)?
You can also say:
- Efter vilan är jag lugn. = After the rest, I am calm.
känna sig
- adjective emphasizes your subjective feeling: “I feel calm.”
  vara
  - adjective states a fact or state: “I am calm.”
In many contexts they overlap, and both are okay. Using känner mig sounds a bit more like you’re describing your personal inner experience.

What exactly is the role of mig in känner jag mig lugn? Why can’t I leave it out?

mig is a reflexive pronoun meaning myself (object form of jag).

The verb here is the reflexive verb känna sig (to feel, literally “to feel oneself”):

Jag känner mig trött. = I feel tired.
Hon känner sig sjuk. = She feels ill.
Vi känner oss glada. = We feel happy.

If you drop mig, the verb changes meaning. känna without a reflexive pronoun usually means:

to know (a person/place): Jag känner henne. = I know her.
to feel/touch: Känn på vattnet! = Feel the water!

So känner jag lugn is wrong, because lugn cannot be the direct object of känna in that way. You need the pattern känna sig + adjective.

Why is it känner and not känns in this sentence?

Swedish has two related but different forms:

känna sig = to feel (reflexive, with a person as subject)
- Jag känner mig lugn. = I feel calm.
kännas (with -s) = to feel (impersonal/passive-like)
- Det känns lugnt. = It feels calm.

So:

Jag känner mig lugn. (I am the one doing the feeling.)
Det känns lugnt. (The situation/it feels calm.)

In Efter vilan känner jag mig lugn, jag is the subject, so you use känner with mig, not känns.

Why is the adjective lugn and not lugnt or lugna?

Here lugn is a predicative adjective describing jag.

With jag, du, han, hon, den, det in the singular, you normally use the base form of the adjective:

Jag är lugn.
Hon känner sig trött.
Han är glad.

You only change form in these cases:

Neuter thing as subject:
- Huset är stort. = The house is big.
- Det är lugnt. = It is calm. (here det refers to a situation, feels like a neuter “it”)
Plural subject:
- Vi är lugna. = We are calm.
- Barnen är trötta. = The children are tired.

In your sentence, the subject is jag (singular), so you use lugn.

What’s the difference between Efter vilan and something like Efter att jag har vilat?

Both express a time relation (“after resting”), but they differ in structure:

Efter vilan
- vila is used as a noun (“the rest/nap/break”).
- Literally: After the rest.
- Short and natural in everyday Swedish.
Efter att jag har vilat / Efter att ha vilat
- vila is used as a verb (“to rest”).
- Literally: After I have rested / After having rested.
- Slightly more explicit about the action.

All of these are grammatically fine:

Efter vilan känner jag mig lugn.
Efter att jag har vilat känner jag mig lugn.
Efter att ha vilat känner jag mig lugn.

The noun phrase Efter vilan is the simplest and probably the most common in casual speech here.

Does vila mean “nap”, “rest” or “break”? Could I use rast or paus instead?

vila is a general word for rest (physically or mentally).

ta en vila / vila = to rest
vilan = the rest (period of rest)

Depending on context, vilan can be understood as:

a nap (especially with children)
a lie-down
a rest period

Related words:

rast = a scheduled break (school break, work break)
- Efter rasten känner jag mig lugn. = After the break I feel calm.
paus = a pause/break in an activity (can be short)
- Efter pausen känner jag mig lugn.

All three can be possible, but they have slightly different nuances:

vila focuses on resting.
rast is more like a fixed break time.
paus is any kind of pause in what you’re doing.

How do you pronounce känner and lugn?

Approximate guidance in English terms:

känner
- kä like “shehn” (short e sound, like in English “ten”, but fronted)
- nn as in “net”
- er like “ehr” (unstressed)
  So roughly: SHEH-nehr (with a short first vowel).
lugn
- lu is like “lugg” but with a shorter u sound (close to “look” but even shorter and more central).
- gn here sounds like “ngn”, similar to “lung” in English, but with a very short final n.
  So roughly: lung (one syllable).

Native-like pronunciation will come with exposure, but these approximations should help you recognise the words when you hear them.