Learn German: Ich frage die Ärztin. - I ask the doctor female.

In German, you usually add the ending -in to a profession to show that the person is female. The male word for doctor is Arzt. To make it feminine, we add -in and also add an umlaut (the two dots) over the A, making it Ärztin.

Is die in this sentence the subject or the object?

Here, die Ärztin is the direct object (the person receiving the action of being asked). In German, this is called the accusative case. For feminine words like Ärztin, the word for "the" stays die in the accusative case, so it looks exactly the same as if it were the subject.

How would I say "I am asking the doctor"?

You would use the exact same sentence: Ich frage die Ärztin. German does not have a separate continuous tense (like "am asking" or "is doing"). The simple present tense ich frage translates to both "I ask" and "I am asking".

Do I need a word for "to", like "I ask to the doctor"?

No, the verb fragen (to ask) takes a direct object right after it, just like in English. You do not need any extra prepositions. You simply say Ich frage followed immediately by the person you are asking.