Hi Ewa, I'm very glad you're learning Mongolian. 

After I have seen your question, I did some research. It seems like there is too little amount of online resources to learn Mongolian. Anyway, I found some things, some of them might already be known to you.

There are some vocabulary courses on memrise. I have chosen 4 from them, which I found the best,  particularly they have audios.

    1. Mongolian 1: https://www.memrise.com/course/1779700/mongolian-1/

    2. Mongolian 2: https://www.memrise.com/course/1779744/mongolian-2/

    3. Mongolian for beginners: https://www.memrise.com/course/1487725/mongolian-for-beginners-mgl123/

    4. Beginner Mongolian: https://www.memrise.com/course/55099/beginner-mongolian/

And then I found this site can be very helpful:  https://www.livelingua.com/project/peace-corps/Mongolian/

and this old book: https://files.eric.ed.gov/fulltext/ED401762.pdf.

Finally, I want you to cast an eye on this Quora question, that is relevant to you.: https://www.quora.com/What-is-the-best-program-for-learning-Mongolian

If you post at some point your written things in Mongolian here, I will try my best to suggest improvements. Feel free :)

Awesome! Thank you Demian so much!

I was thinking about it as I want to learn a lot about Mongolian history.

Maybe in 2-3 years I will be able to speak Mongolian fluently and than I can start with traditional Mongolian script. 

I cannot wait!

I really appreciate your feedback. It made my day. :)