Om jeg så er træt efter arbejdet, går jeg først ind, når jeg har vandet planterne og lukket buret.

Yes, om often means about or introduces an indirect if/whether clause, but here it is part of the concessive pattern om ... så ....

In this sentence:

• Om jeg så er træt efter arbejdet
  means something like:
• Even if I am tired after work
• Although I may be tired after work

So this is not the usual if/whether meaning. It is an idiomatic structure showing contrast: being tired does not change what happens next.

In om ... så ..., så helps create the meaning even if / no matter that.

So:

• om jeg er træt would not sound like the same idiomatic concessive structure
• om jeg så er træt gives the sense even if I am tired

In many cases, så is not translated word-for-word. It is there to strengthen the expression and make the clause sound natural in Danish.

Because Danish main clauses normally follow the verb-second rule.

The sentence begins with a fronted subordinate clause:

• Om jeg så er træt efter arbejdet

After that, the main clause starts, and in a Danish main clause the finite verb comes in second position:

• går jeg først ind

So the order is:

1. fronted clause
2. finite verb
3. subject

That is why you get:

• Om jeg så er træt efter arbejdet, går jeg først ind ...

and not:

• Om jeg så er træt efter arbejdet, jeg går først ind ...

Gå ind means go inside or go in.

Danish often uses a verb plus a directional particle:

• gå ind = go in / go inside
• gå ud = go out
• komme hjem = come home

In a simple main clause, the particle often stays after the verb:

• jeg går ind
• han går ud

So går ind is completely normal. It works much like English go in.

Here først means only then, not until then, or only after that.

So:

• går jeg først ind, når ...

has the sense:

• I only go inside when/once ...
• I do not go inside until ...

This is a very common use of først in Danish. It does not mean first in a numbered sequence here. It means that the action of going inside happens only after the other actions are completed.

Because når is the normal word for when in a clause about something that happens regularly, generally, or in the future.

Here the meaning is something like:

• I only go inside when I have watered the plants and closed the cage

That is a general or habitual situation, so når fits well.

Compare:

• når = when, whenever, once
• da = when, but usually about a specific event in the past
• hvis = if

So:

• når jeg har vandet planterne ... = when/once I have watered the plants ...

Because Danish often uses the present perfect to show that one action must be completed before another action happens.

Here the sequence is:

1. water the plants
2. close the cage
3. go inside

So Danish says:

• når jeg har vandet planterne og lukket buret

literally:

• when I have watered the plants and closed the cage

This is very similar to English, which can also say:

• when I have finished
• once I have done it

The perfect form emphasizes completion before the main action.

Because one auxiliary verb can govern both past participles.

So:

• har vandet planterne og lukket buret

means:

• have watered the plants and closed the cage

This is exactly like English:

• I have watered the plants and closed the cage

You do not need to repeat har unless you want extra emphasis or a different structure.

Because they refer to specific things:

• planterne = the plants
• buret = the cage

In Danish, the definite article is usually attached to the noun:

• plante → planten / planterne
• bur → buret

So the speaker is talking about particular plants and a particular cage that are already known in the situation.

Efter arbejdet means after work or more literally after the work/the workday.

The form arbejdet is the definite form of arbejde:

• et arbejde = a job / a piece of work
• arbejdet = the work

In everyday Danish, efter arbejdet is a natural way to say after work. It often refers to the speaker’s workday rather than a specific task.

Because træt is an adjective, and Danish uses være with adjectives describing a state.

So:

• jeg er træt = I am tired

This is like English. Danish does not say I have tired for this meaning.

Other similar examples:

• jeg er sulten = I am hungry
• jeg er glad = I am happy
• jeg er færdig = I am finished

Those commas mark clause boundaries.

• Om jeg så er træt efter arbejdet,
  this is a subordinate clause placed before the main clause

• går jeg først ind, når jeg har vandet planterne og lukket buret
  here når starts another subordinate clause

So the commas help show the structure:

1. concessive subordinate clause
2. main clause
3. temporal subordinate clause

In written Danish, commas are heavily tied to clause structure, so this punctuation is very typical.