There are classifiers in several languages (e.g. Chinese), also known as measure words, for each noun. I think that it is a good idea to start learning them when I do not yet know many nouns, so that that won’t be a burden later. On the other hand, this will slow down the process of learning the language significantly and I heard that even native speakers sometimes have to use a special dictionary for looking up classifiers.
It is important to distinguish between classifiers and measure words. Though they are commonly referred to as synonyms, their uses are distinct. Some classifiers change the meaning of the noun they are attached to, and are not just for counting. I will use the word classifier as it encompasses your question better.
Outside of that, let me use the example of Chinese to illustrate. Using the generic ge (个) for classifying nouns is what most Chinese (Taiwanese in my case) people recommend when beginning. This can be used for just about everything. As you start to learn what each classifier implies (本 for bound books, magazines, &c; 张 for flat things like paper, newspaper, &c.; 封 for letters and things within envelopes), you can add more classifiers to your common use.
With that said, when learning basic nouns and noun categories, it is useful to learn the most common classifiers when learning nouns. However, many native speakers themselves may ignore the "correct" classifier to use the generic classifier if convenience serves.
Classifiers are sometimes compared to grammatical gender of other languages (though they are not the same) because of the classification given to a class of nouns. However, they are distinct from gender because some nouns may fall into multiple classes whereas grammatical gender is an essential (and virtually unchanging) aspect of the noun.
One benefit of learning the classifiers commonly associated with a noun group is that you do not need to go back to relearn them or learn additional ones. However, this is only a slight advantage as many classifier-using languages have intricacies in the usage of different classifiers and even native speakers may disagree on the "proper" usage. Waiting until you have a better understanding of the language will help you understand and grasp the proper use cases of the unusual classifiers. (I remember in my Advanced Chinese Linguistics class having an entire unit on classifiers and the teacher was offering a class the next semester solely on classifiers and modifiers in Chinese. So, this is a very involved subject.)
This list of Chinese Classifiers may be useful for understanding classifiers in the Chinese language, though the principle of categorization of nouns is not unique to Chinese (or Afro-Asiatic Languages for that matter).
I would learn the most common classifiers with the words as soon as I can. That's because most nouns are "matched" to classifiers. Some nouns also have "esoteric" classifiers that can be learned later.
Not doing so would be like learning nouns without their genders in most European languages. That's possible of course. But it's more trouble to "go back" and learn the proper gender for each noun "der Hund," "das Haus," etc. than to learn them in together with the noun the first place.