Learn Hardcore Russian: Сын приходит домой весёлый после тренировки в спортзале. - My son comes home cheerful after training at the gym.

идти / идёт – to go (on foot), in one direction, with focus on the process:
- Сын идёт домой. – The son is going home (now, on his way).
ходить / ходит – to go (on foot) habitually or back-and-forth:
- Сын ходит домой пешком. – The son goes home on foot (as a habit).
приходить / приходит – to come, to arrive (imperfective); focus is on the arrival at a place:
- Сын приходит домой. – The son comes home (arrives home).

In Сын приходит домой весёлый после тренировки в спортзале, the idea is “he (regularly) comes home from training,” i.e., he reaches home after the training. That’s why we use the verb with the prefix при-, which adds the idea of arriving.

Does приходит mean “comes” or “is coming”?

It can mean either, depending on context. Russian present tense covers both English Present Simple and Present Continuous.

Habitual/repeated action:
- Сын всегда приходит домой весёлый.
  → The son always comes home cheerful.
Action in progress / right now, in a narrative:
- Слышу, дверь открывается: сын приходит домой.
  → I hear the door open: the son is coming home.

In your sentence, without extra context, the most natural reading is habitual:
Сын приходит домой весёлый после тренировки в спортзале.
→ The son (usually) comes home cheerful after training at the gym.

Why is it домой and not дом or дома?

These are different forms with different meanings:

дом – house, home (nominative, as a subject or object)
- Дом большой. – The house is big.
- Я вижу дом. – I see a house.
дома – at home (location)
- Сын дома. – The son is at home.
домой – (to) home (direction, movement towards home)
- Сын идёт домой. – The son is going home.

In your sentence we have motion to home (he comes home), so the directional form домой is required:
Сын приходит домой… – The son comes home…

What does весёлый do in this sentence, and what does it agree with?

Весёлый is an adjective meaning cheerful, in a good mood.

Here it stands in nominative masculine singular and agrees with сын (also nominative masculine singular). It functions like a “secondary predicate” – it describes the state of the son when he comes home:

Сын приходит домой весёлый.
Literally: The son comes home (being) cheerful.

So grammatically:

сын – subject (nominative)
приходит – verb (3rd person singular)
весёлый – adjective in nominative, agreeing with сын, describing his state.

Could I say Сын приходит домой весёлым instead of весёлый?

Yes, весёлым (instrumental case) is also possible, and the meaning is very close, but there is a nuance:

Сын приходит домой весёлый. (nominative)
Slightly more like a resulting or evident state: He comes home, and he is cheerful.
Сын приходит домой весёлым. (instrumental)
More like a state during the action: He comes home in a cheerful mood / in a cheerful state.

In everyday speech many speakers wouldn’t feel a strong difference here, but:

Nominative (весёлый) is very common with verbs like приходить, возвращаться to describe what he “is” when he appears.
Instrumental (весёлым) tends to emphasize the “role” or “mode” in which he comes.

Both are grammatically correct.

Why is it весёлый and not the adverb весело?

весёлый – adjective: describes a person’s characteristic or emotional state (cheerful).
весело – adverb or short-form predicative: can mean cheerfully or it is fun / cheerful.

Compare:

Сын приходит домой весёлый.
→ The son comes home cheerful (he himself is cheerful).
Сын приходит домой весело.
→ The son comes home cheerfully (in a cheerful manner). – This describes how he comes, not what mood he is in as a state.

In your sentence, the focus is on his mood, so the adjective весёлый is more natural.

Why is it после тренировки, with тренировки in the genitive?

The preposition после (after) always requires the genitive case.

после чего? – after what? → genitive
тренировка (nom.) → тренировки (gen. sing.)

So:

после тренировки – after (the) training / practice session

Other examples:

после работы – after work
после обеда – after lunch

If you wanted to say after training sessions (in general), you could use the plural genitive:

после тренировок – after trainings (plural, general).

Does в спортзале belong to тренировки or to приходит домой?

In Сын приходит домой весёлый после тренировки в спортзале, the most natural reading is:

после тренировки в спортзале – after a training session in the gym

So в спортзале describes где? the training took place.

If we wanted в спортзале to describe where he comes home, that would sound strange logically: “comes home in the gym”. So by meaning and normal interpretation, в спортзале attaches to тренировки.

You can make it clearer by reordering:

Сын после тренировки в спортзале приходит домой весёлый.

What’s the difference between в спортзале and в спортзал?

They differ in case and therefore in meaning:

в спортзале – prepositional case: in the gym (location)
- Он тренируется в спортзале. – He trains in the gym.
в спортзал – accusative case: to the gym (direction)
- Он идёт в спортзал. – He is going to the gym.

In your sentence we are talking about training in the gym, not going to the gym, so в спортзале (in the gym) is correct:
после тренировки в спортзале – after training in the gym.

Is the word спортзал one word or two? What does it literally mean?

Спортзал is written as one word. It is a shortened compound from:

спортивный зал – sports hall / gymnasium

Russian often fuses such combinations into one word in informal or semi-formal style:

спортзал (спортивный зал) – gym
спорттовары (спортивные товары) – sports goods

So в спортзале literally is in the sports hall, idiomatically in the gym.

How flexible is the word order in this sentence? Can I move parts around?

Russian word order is relatively flexible. You can move adverbials (like после тренировки в спортзале, домой, весёлый) to change emphasis.

All of these are grammatically possible, with slightly different focus:

Сын приходит домой весёлый после тренировки в спортзале.
– Neutral: the son comes home cheerful after training in the gym.
Сын после тренировки в спортзале приходит домой весёлый.
– Emphasis on after training in the gym: that’s the time when this happens.
После тренировки в спортзале сын приходит домой весёлый.
– Very natural, emphasizes the time setting first: After training in the gym, the son comes home cheerful.
Сын приходит домой после тренировки в спортзале весёлый.
– Also possible, though the long tail at the end is less elegant in speech.

What you generally don’t change:

Сын as subject near the beginning.
приходит as the main verb following the subject.

Everything else is more movable.

Why is сын in the nominative case?

Сын (son) is the subject of the sentence, the one who performs the action of приходит (comes).

In Russian, the subject of a finite verb appears in the nominative case. So:

Кто приходит? – Who comes? → сын.

Other examples:

Дочь читает книгу. – Дочь (nom.)
Учитель объясняет правило. – Учитель (nom.)

How would the meaning change if we used придёт instead of приходит?

приходит – imperfective, present tense:
- habitual / repeated: he (usually) comes
- can be used in a narrative as current action
придёт – perfective, future tense:
- one-time, completed arrival: he will come (once, at a specific time).

So:

Сын приходит домой весёлый после тренировки в спортзале.
→ The son (usually) comes home cheerful after training at the gym.
Сын придёт домой весёлый после тренировки в спортзале.
→ The son will come home cheerful after the (upcoming) training at the gym.

The second one talks about a future event, not a regular habit.

Is тренировка only “sports training,” or can it mean other kinds of practice?

Тренировка most often refers to physical / sports training, a practice session:

тренировка по футболу – football practice
утренняя тренировка – morning workout

It can also be used a bit more broadly for “training” or “drill,” but usually with some physical or practical aspect:

огневая тренировка – firearms training
тренировка памяти – memory training (more figurative/extended use)

In your sentence, because of в спортзале, it clearly means a sports training session in the gym.