Learn Farsi: من می‌خواهم زبان فارسی یاد بگیرم.

In Farsi, when you say you "want to do" something, the second verb changes to the subjunctive form. This usually means replacing the continuous prefix میـ with the subjunctive prefix بـ. So, instead of می‌گیرم (I get/take), we use بگیرم.

The dictionary form of "to learn" is یاد گرفتن. How does the subjunctive work with two-part verbs?

For compound verbs like یاد گرفتن, the first word (یاد) stays exactly the same. You only add the subjunctive prefix بـ to the actual action verb part (گرفتن). That is why it becomes یاد بگیرم.

Is there a word for "to" in this sentence, like in the English "want to learn"?

No, Farsi doesn't use a linking word like "to". The first verb (می‌خواهم, I want) simply triggers the subjunctive form in the second verb (یاد بگیرم, learn), which is placed at the very end of the sentence.

How do I pronounce زبان فارسی together?

It is pronounced zabân-e fârsi. There is a hidden "e" sound connecting the two words. This is called the Ezafe, and it acts as a bridge to link a noun (زبان, language) to its describing adjective (فارسی, Persian).