What about the following:
"Beszélj velem magyarul, próbálok megtanulni (= próbálok megtanulni beszélni magyarul)"
It is a perfect solution.
Just some more thoughts in italic, due to the links shared it is mainly for Hungarian natives. As a disclaimer I am not fully aware of the English grammar terms, so I looked up for those ("szófaj"/"parts of speech" and "mondatrész"/"constituent") in wikipedia/dictionaries.
"
In Hungarian" is definitely a noun in English. And "magyarul" is definitely NOT a noun in Hungarian. But what is it in fact?
I feel "magyarul" is not a simple adverb, but more an adjective (as a part of speech). Nevertheless in the sentences above "magyarul" is a constituent with an adverbial function.
My statement is based on the examples here, Állapothatározó – Wikipédia. Certainly "magyarul" is not an "állapothatározó", but seems to be a sort of "módhatározó" in this case.
Or look for "rendületlenül" here A HATÁROZÓK.
There is a similiarity with words ending not with -ul/-ül, but with -an/-en.
See also -an, -en ragos melléknév.
As a bonus see also here Hol ford -ul/-ül elő?
It is quite a complicated matter for non-linguists, eg. I am not at all.
Finally let's try to understand why the solution from Francisgranada was perfect.
If only "próbáloK megtanulni" is used and it is omitted to express what it is in fact that I am trying to learn, then it is quite comfortable. But let's continue it in different ways and test our ears and mind:
"Beszélj velem magyarul, próbáloK megtanulni VALAMIT" - something/object
"Beszélj velem magyarul, próbáloK megtanulni VALAMILYEN MÓDON" - somehow/adverbial function
Both seem to be fine for me. So it is still comfortable to use "próbálok", which is "alanyi igeragozás" / "verb conjugation referring to an undefined/general object"
Not let's get back to Bilmemki's versions, in other words to "próbálom", which is "tárgyas igeragozás"/ "verb conjugation with a focus on a defined object"
"Beszélj velem magyarul, próbálom megtanulni AZT /AZT A DOLGOT" - that thing/object
To be honest it does not sound good, my mind does not accept "magyarul" as an object.
So in case of difficulties, just simply use "alanyi ragozás".
