Yau zan tafi gida in ga iyali na.

Yau means today.
It is a time adverb, and in Hausa it very often appears at the beginning of the sentence for emphasis or clarity:

• Yau zan tafi gida… – Today I will go home…

You can also move it later:

• Zan tafi gida yau in ga iyali na.

The meaning is the same; putting Yau first just emphasizes today a bit more, like in English: Today, I’m going home…

zan means I will and marks the future tense for the first person singular.

It comes from two parts:

• za – future marker (will / going to)
• ni – I / me

Spoken together, za ni becomes zan.
Compare with other persons:

• za ka tafi – you (m.sg.) will go
• za mu tafi – we will go

So zan tafi literally is will‑I go → I will go.

Both tafi and je can translate as to go, but there are some tendencies:

• tafi often has the idea of leaving / departing / setting off
  
  • Zan tafi gida. – I will go (leave) home.
• je is a more neutral go (to a place)
  
  • Zan je kasuwa. – I will go to the market.

In this sentence, Zan tafi gida is very natural, because going home often feels like “leaving” where you are now.
You could also hear Zan je gida, and it would still be understood as I will go home, just a bit more neutral.

In Hausa, many verbs of motion don’t need a preposition before a destination.
So gida (home) can come directly after tafi:

• Zan tafi gida. – I will go home.
• Zan tafi Kano. – I will go to Kano.

You can say zuwa gida (“to home”), but in this very common pattern, Hausa usually just uses the bare noun:

• English needs to: go to home
• Hausa doesn’t: tafi gida

in ga here means roughly to see or in order to see.

• in is a little conjunction that originally means if / when, but in this kind of sentence it often works like (and) to / so that (I) can.
• ga is the verb see in the future/subjunctive form (compare: zan ga shi – I will see him).

Because the subject I is the same as in the first clause (zan tafi), Hausa does not repeat the pronoun:

• Yau zan tafi gida in ga iyali na.
  → Today I will go home *to see my family.*

So you can think of in ga here as a compact way of saying so that I see / so that I can see.

It can be repeated, but it doesn’t have to be when the subject stays the same.

Full form (more explicit):

• Yau zan tafi gida in na ga iyali na.
  – Today I will go home if/so that I see my family.

Common, shorter form (same meaning here):

• Yau zan tafi gida in ga iyali na.

In everyday speech and writing, when the second verb has the same subject as the first one, Hausa often leaves out the repeated pronoun after in. That’s what is happening in the original sentence.

They are closely related:

• ga is the finite verb stem used with tense markers:
  
  • Na ga shi. – I saw him.
  • Zan ga ita. – I will see her.
• gani is the verbal noun seeing and also acts as an infinitive-like form:
  
  • Ina son ganin ka. – I like seeing you / I want to see you.

In in ga iyali na, ga is the verb see in a purpose/conditional clause introduced by in.
You couldn’t put gani there; in gani would be ungrammatical in this pattern.

Hausa usually puts the possessed noun first, then the possessive pronoun after it:

• iyali na – family my → my family
• mota ta – car her → my car (for 1st person fem. forms in some dialects/writing)
• littafi na – book my → my book

So the order is: thing + possessor pronoun.
na iyali would mean of the family, not my family.

Also, in writing you often see the pronoun attached:

• iyalina = iyali na

Both iyali na and iyalina mean my family.

Yes, but the meaning changes slightly:

• Zan tafi gida – I will go home (destination)
• Ina gida – I am at home (location)
• Zan zauna a gida – I will stay at home

With tafi (go), gida by itself normally shows the destination (going to home), so a is unnecessary and would sound odd there. a gida is more about being at home than going home.

In modern standard Hausa, the most natural form is the combined one: zan tafi.

Historically/underlyingly:

• za ni tafi → zan tafi

In normal usage:

• zan tafi gida – correct and natural
• za ni tafi gida – possible in very slow, careful speech, but sounds unnatural or overly separated in most contexts.

So you should learn and use zan, za ka, za mu, etc., as fixed future forms.

zan tafi is a future form, but in context it often overlaps with English I’m going (to), especially for plans later the same day:

• Yau zan tafi gida…
  – Today I will go home.
  – Today I’m going home.

It usually implies an intention or plan, not something happening right this second.
For something happening now, Hausa would more typically use a different aspect, e.g.:

• Ina tafiya gida yanzu. – I’m going home now (right now, in progress).

Yes, that is natural and grammatical:

• Yau zan tafi gida in ga iyali na.
• Zan tafi gida yau in ga iyali na.

Both mean the same thing.
Hausa is fairly flexible with where it places time expressions like yau. Putting yau at the beginning makes today a bit more prominent, but both orders are common and correct.