!سلام

.من فارسی یاد می گیرم
.من انگلیسی بلدم و من اهل انگلیس هستم

some important points:

Be carefull in using pronouns in persian because in persian verbs you have to use pronoun then using it again in sentence in general is not correct and it calls حشو which means extra

but it is used in some cases for emphasize on yourself which is not polite in all the situation specialy in case of talking about yourself so it is better to eliminate all من pronouns because from the م which called ضمیر چسبان or close-fitting pronoun

but about sentences timing if you said فارسی یاد می گیرم you are talking about the moment you are doing this but if you want to say about somthing which is more continous and start in past and may continue in future say  دارم فارسی یاد میگیرم or  در حال یادگیری زبان فارسی هستم the difference is in their application دارم فارسی یاد می گیرم is informal and used just in speech also its time is shorter but درحال یادگیری زبان فارسی هستم is more formal and also used in speech and the time of doing is more longer.

but the most beautiful sentence to tell this is: مشغول یادگیری زبان فارسی هستم

this sentence is formal and also common use in speech it is also mention to jobs which need time like language learning. or another example is about education if someone ask you رشته تحصیلی شما چیست؟ what is your education field? it is better to answer مشغول تحصیل در رشته مهندسی هستم I am educating in engineering field.

Ballboy explanation about two other sentences is complete.