Dimarts no treballo, així que aniré a la biblioteca.

In Catalan, days of the week are often used without a preposition when they express time.

So:

• Dimarts = Tuesday / on Tuesday
• Dilluns = Monday / on Monday

That is very normal Catalan.

A useful contrast:

• Dimarts no treballo = On Tuesday, I’m not working / Tuesday I don’t work
• Els dimarts no treballo = I don’t work on Tuesdays (habitual, every Tuesday)

So in your sentence, Dimarts refers to a specific Tuesday, not a repeated habit.

Yes — in Catalan, weekdays are normally lowercase:

• dimarts
• dimecres
• dijous

Here it is capitalized only because it is the first word of the sentence. If it appeared in the middle, you would usually write:

• No treballo dimarts.

So Catalan is like English at the start of a sentence, but unlike English in general usage, because weekdays are not usually capitalized.

Because Catalan often drops subject pronouns when the verb form already shows who the subject is.

• treballo already means I work
• the -o ending tells you it is first person singular

So:

• (Jo) treballo = I work

Both are possible, but jo is usually omitted unless you want emphasis, contrast, or clarity.

For example:

• Jo no treballo, però ell sí. = I’m not working, but he is.

In your sentence, jo is unnecessary.

Treballo is the 1st person singular present indicative of treballar = to work.

The verb is conjugated like this in the present:

• jo treballo = I work
• tu treballes = you work
• ell/ella treballa = he/she works
• nosaltres treballem = we work
• vosaltres treballeu = you all work
• ells/elles treballen = they work

So no treballo means I do not work / I’m not working depending on context.

Because standard Catalan negation is formed with no + verb.

So:

• treballo = I work
• no treballo = I don’t work

This is the normal position. English uses do not work, but Catalan does not need an extra auxiliary like do.

A few examples:

• No menjo carn. = I don’t eat meat.
• No vindré demà. = I won’t come tomorrow.

So no treballo is exactly the expected Catalan word order.

Així que means so, therefore, or so then. It introduces a result or consequence.

So the logic is:

• Dimarts no treballo = cause/situation
• així que aniré a la biblioteca = result

It is not the same as because.

Compare:

• No treballo, així que aniré a la biblioteca.
  = I’m not working, so I’ll go to the library.

• Aniré a la biblioteca perquè no treballo.
  = I’ll go to the library because I’m not working.

So:

• així que = result
• perquè = cause/explanation

Aniré is the 1st person singular future of anar = to go.

It means:

• aniré = I will go

The future of anar is irregular, so it does not simply look like the infinitive plus an ending in the most obvious way. The future forms are:

• aniré = I will go
• aniràs = you will go
• anirà = he/she will go
• anirem = we will go
• anireu = you all will go
• aniran = they will go

So in your sentence, aniré a la biblioteca means I’ll go to the library.

Yes, sometimes Catalan uses the present tense to talk about the near future, especially when there is a clear time expression.

For example:

• Dimarts vaig a la biblioteca. can mean On Tuesday I’m going to the library.

However, aniré is clearer here because it explicitly marks the future:

• aniré = I will go

That makes the sentence sound very natural as a decision or consequence:

• I’m not working Tuesday, so I’ll go to the library.

Using the future here avoids possible ambiguity and sounds very straightforward.

A is the preposition meaning to, and la biblioteca means the library.

So:

• a la biblioteca = to the library

The article is used because Catalan normally says the library, not just library, in this kind of sentence.

As for contraction:

• a + el → al
• a + els → als

But:

• a + la stays a la
• a + les stays a les

Since biblioteca is feminine singular, you get:

• a la biblioteca

Compare:

• Vaig al banc. = I’m going to the bank.
• Vaig a la biblioteca. = I’m going to the library.

The accents mainly help show which syllable is stressed, and in some cases they also indicate vowel quality.

• així is stressed on the last syllable: ai-xí
• aniré is stressed on the last syllable: a-ni-ré

Very rough English approximations:

• així ≈ eye-SHEE or uhy-SHEE depending on accent
• aniré ≈ ah-nee-REH

A few helpful pronunciation notes:

• the x in així sounds like sh
• the accented í tells you the stress falls there
• the accented é tells you the stress falls there, and that it is a closed e sound

So the accents are not optional decoration — they are part of the correct written form and help you pronounce the words properly.