How to download itunes to my iphone?

How do I download itunes to my iphone?

It's already on there.

All apple products will come with an itunes app installed.
If you are missing one then maybe you have turned it off in restrictions

Go to settings-->general-->restrictions
If you have it turned on then check to make sure you have "itunes" unchecked for being restricted.

Once you have received an iPhone, certain applications are already installed, such as Messages, Calculator, Phone, Mail, Safari, and of course iTunes. ITunes is where you can download new hit songs onto your phone so you can take your favorite music on the go.
To get songs from iTunes and downloaded to your mobile device, you need to have a iTunes gift card, and you then need to redeem your card on iTunes so you have a balance to buy all the songs you want.

To buy iTunes gift cards, you can go to almost any store such as: Apple stores, Walmart, Safeway, Walgreens, and more.
iTunes is already installed nito your iPhone along with other apps.

Appstore icon is already there

The reason I'm asking is that I was told I had to use it to add my music to my phone.

You should download and install iTunes on your computer, and then use it to put music from computer to iPhone, not download iTunes to your iPhone. If you want to know how to convert video and put DVD movie on iPhone/iPod, I think this step by step guide would help you.