Azi am o zi liberă și vreau să vizitez familia mea.

Both azi and astăzi mean today.

• Azi is shorter and more informal, very common in everyday speech.
• Astăzi is slightly more formal or neutral; you’ll hear it in news, writing, or careful speech.

In this sentence you can use either:

• Azi am o zi liberă...
• Astăzi am o zi liberă...

Both are correct and mean the same thing.

Both structures exist, but they’re not identical in meaning.

• Am o zi liberă = I have a day off
  Literally: I have a free day.
  Focus: you are talking about a specific day that is free (e.g., from work or school).

• Sunt liber / Sunt liberă = I am free
  Focus: your current state (you’re not busy), not necessarily that this is an officially free day.

In the context of planning to visit family, am o zi liberă sounds natural because you’re referring to a particular day off.

In Romanian, articles and adjectives must agree in gender and number with the noun.

• zi (day) is feminine singular.
• The indefinite article for feminine singular is o (not un).
• The adjective liber must also be feminine singular: liberă.

So:

• o zi liberă = a free day
  (o = fem. article, zi = fem. noun, liberă = fem. adjective)

If the noun were masculine, it would be:

• un băiat liber (a free boy)

Am is the 1st person singular (I) form of the verb a avea (to have) in the present tense.

• a avea = to have
  
  • eu am = I have
  • tu ai = you have
  • el/ea are = he/she has
  • noi avem = we have
  • voi aveți = you (pl.) have
  • ei/ele au = they have

So Azi am o zi liberă = Today I have a free day / a day off.

Romanian does not usually use the bare infinitive after vreau (I want).
Instead, it uses the subjunctive with să:

• vreau să vizitez = I want to visit
• literally: I want that I (should) visit

Patterns:

• vreau să merg = I want to go
• vreau să mănânc = I want to eat
• vreau să citesc = I want to read

So after vreau, the natural structure is vreau să + [verb in 1st person singular subjunctive], not an infinitive like a vizita.

• a vizita = to visit (the infinitive)
• vizitez = 1st person singular, present tense (I visit)

In the sentence:

• vreau să vizitez = I want to visit

Here, vizitez is the present tense form used with să (subjunctive), but it looks the same as the normal present indicative 1st person singular.

Conjugation of a vizita (to visit), present tense:

• eu vizitez = I visit
• tu vizitezi = you visit
• el/ea vizitează = he/she visits
• noi vizităm = we visit
• voi vizitați = you (pl.) visit
• ei/ele vizitează = they visit

Several things are happening here:

1. familie (family) is the base noun (feminine).
2. With the definite article, it becomes familia = the family.
3. The possessive mea = my, feminine singular.

In Romanian, the normal way to say my family is:

• familia mea = the family of mine → my family

Not:

• familie mea → incorrect (the noun needs its article here)
• familia alone would mean the family, without specifying my.

So vreau să vizitez familia mea = I want to visit my family.

The form of my depends on the gender and number of the noun it modifies.

For singular nouns:

• Masculine/neuter: meu
• Feminine: mea

Since familia (family) is feminine singular, you must use mea:

• familia mea = my family

Examples:

• fratele meu (my brother – masculine)
• sora mea (my sister – feminine)
• telefonul meu (my phone – masculine)
• mașina mea (my car – feminine)

Yes, that’s very natural and actually more common in everyday speech:

• Vreau să-mi vizitez familia.
  Literally: I want to visit my family
  Here, mi is a clitic pronoun meaning my, attached to vizitez.

Both are correct:

• vreau să vizitez familia mea – a bit more explicit, slightly more formal or emphatic on my.
• vreau să-mi vizitez familia – very idiomatic, smooth Romanian.

Romanian has some flexibility, but not all orders sound natural. Some acceptable variants:

• Azi am o zi liberă și vreau să-mi vizitez familia.
• Astăzi am o zi liberă și vreau să vizitez familia mea.
• Am azi o zi liberă și vreau să vizitez familia mea. (still okay; slight emphasis on azi)

Less natural or wrong:

• Azi o zi liberă am și vreau să vizitez familia mea. (awkward)
• Vreau să vizitez mea familia. (incorrect order; the possessive follows the noun and usually its article)

The original word order is the default, most natural version.