"I would really like to visit East Asia."
Translation:J'aimerais beaucoup visiter l'Asie de l'est.
Except with "en", which acts as a determiner in such cases, so no article is required.
And obviously (I hope), "au" and "du" already include a definite article, so no second article is required. This is also true of "aux" and "des" (in its "de + les = des" form), which is relevant to the USA and a few other plural states.
It's not grammatically wrong, but it is not likely to be appropriate in this context because it indicates an intent. You would like to do it so strongly that you intend to do it.
If you're discussing a trip with a travel agent, then it would be correct.
Don't forget that the literal meaning of "je voudrais" is "I would want". Yet in English, we only ever express it that way in the negative. In the affirmative, "I would like" does double-duty.