Merhaba, 

Here is part of a song by my favorite band: 

I am not sure how the suffix "ığım" in "başladığım" is made, when it is used, and what it means. This is just one exmple that I could find for now but I hear it A LOT in the movies and songs. I would really appreciate if someone could please explain it to me.

Teşekkürlar 

...Başladığım gün: the day that I started..

Here "başladığım" is an adjective which is derived from the verb "başlamak".

başla-dı-ğım: that I started.

başla-dı-ğın: that you started.

başla-y-acağ-ım: that I will start.

başla-y-ama-dı-ğım: that I couldn´t start.

You can produce adjectives, adverbs, pronouns from a verb. "-dık" is the suffix that makes adjectives. For example : "Tanıdık insan".

...Başladığım gün: the day that I started..

Here "başladığım" is an adjective which is derived from the verb "başlamak".

başla-dı-ğım: that I started.

başla-dı-ğın: that you started.

başla-y-acağ-ım: that I will start.

başla-y-ama-dı-ğım: that I couldn´t start.

You can produce adjectives, adverbs, pronouns from a verb. "-dık" is the suffix that makes adjectives. For example : "Tanıdık insan".

I don´t think I can give a good explanation but it´s like when you have two verbs, you may need to add ´-dik´ to the verb representing the object. 

I heard her singing.

Onun şarkı söylediğini duydum.

In a way, English is easier because you don´t have to worry about so many endings, just add ´-ing´. But English is also confusing because ´-ing´ is often used for something totally different - present continuous.

Here is like adding possession to a verb:

I know you like coffee. (I know of your liking for coffee)

Kahveyi sevdiğini biliyorum.

Here´s a link: http://www.turkishlanguage.co.uk/objectpart.htm

Çok teşekkür ederim arkadaşlar!!!

Shouldn´t this be kahve sevdiğini biliyorum? (Just ´coffee´, not ´the coffee´

I think either way is OK, not sure which one is better. 

Tunci wrote here http://www.turkishclass.com/forumTitle_58960: 

Sometimes people  drop object marker when they use “sevmek”