Hankin uuden takin.

It is built into the verb hankin.

The ending -n tells you the subject is I / minä. So:

• hankin = I get / I acquire / I buy or I got / I acquired
• Minä hankin uuden takin. is also possible, but minä is often left out because the verb already shows the person.

The dictionary form is hankkia.

That is the infinitive, the form you would look up in a dictionary.

A few related forms:

• hankkia = to get, acquire, obtain, buy
• hankin = I get / I got
• hankit = you get / you got
• hankkii = he/she gets / got

It can be either, depending on context.

This is one of those Finnish verb forms where the 1st person singular present and 1st person singular past look the same.

So Hankin uuden takin. can mean:

• I get/am getting a new coat
• I got a new coat

Context usually makes it clear. For example:

• Huomenna hankin uuden takin. = Tomorrow I’ll get a new coat.
• Eilen hankin uuden takin. = Yesterday I got a new coat.

Not always.

Hankkia is broader than ostaa. It means something like:

• to get
• to acquire
• to obtain
• sometimes to buy

So in this sentence, English might translate it as I got a new coat or I bought a new coat, depending on the situation.

If you want to be very specifically buy, Finnish often uses ostaa:

• Ostin uuden takin. = I bought a new coat.

Because uuden takin is the object of the verb, not the subject.

The basic dictionary forms are:

• uusi = new
• takki = coat

But after the verb hankkia, the thing being obtained is the object, and here Finnish uses the total object form. In the singular, that usually looks like the genitive form.

So:

• uusi takki = a new coat / the new coat as a basic noun phrase
• hankin uuden takin = I got/bought a new coat

Formally, takin is the genitive singular form of takki.

In this sentence, it functions as a total object. Many Finnish grammars explain it like this:

• the object is complete/whole
• the action is seen as reaching its result
• therefore the singular object appears in a genitive-looking form

So takin is genitive in form, but its role in the sentence is object.

A useful contrast:

• Hankin uuden takin. = I got/bought a whole new coat.
• En hankkinut uutta takkia. = I did not get/buy a new coat.
  In a negative sentence, the object goes into the partitive: uutta takkia.

Because adjectives agree with the nouns they describe in Finnish.

So if the noun changes case, the adjective changes too.

Here:

• takki → takin
• uusi → uuden

Both are singular, and both are in the same case form.

This is very normal in Finnish:

• iso talo = a big house
• ison talon = of a big house / a big house as a total object

That is just how this adjective declines. Uusi is one of the common words whose stem changes in oblique cases.

Important forms are:

• uusi = new
• uuden = of a new / a new (object form here)
• uutta = new (partitive)

So you should learn it as a declension pattern:

• uusi, uuden, uutta

This same pattern appears in other similar words, so it is worth memorizing early.

Because of consonant gradation.

The word takki has kk in the basic form, but in some other forms that strong grade becomes a weaker grade:

• takki
• takin
• takissa

So kk → k here.

This is a very common feature in Finnish and affects many nouns and verbs.

Finnish does not have articles like English a/an and the.

So uuden takin could correspond to:

• a new coat
• the new coat

Context decides which English translation sounds right.

Out of context, English usually translates this sentence as I got a new coat, because that is the most natural generic reading.

Yes, it is normal.

What is happening is that the subject pronoun minä has been omitted:

• Minä hankin uuden takin. = full version
• Hankin uuden takin. = very natural shorter version

Because hankin already tells you the subject is I, Finnish often leaves out the pronoun. So the sentence may look like it begins with the verb, but really it is just a normal sentence with an omitted subject.