Ask Steve - Number Systems
I don't teach numbers or letter names. It is more important to be able to speak and read a language first. Now if you are at a high level of fluency after about 1-2 years then I would focus on numbers and letter names. But if you need a high fluency of numbers or letters for your job then use a few flash cards and identify the most frequently used numbers in your target language. I have learned numbers but just the numbers that I need the most and the ones that are printed on money or are associated with coins.
I agree with Ftornay. but also I don't really see why you would make an exception for numbers. You can always do flashcards on a certain area to make some quick progress in that area, but why would numbers be the exception - you could argue the same for colors/names of trees,flowers/food and ingredients etc.
The pronounciation in itself doesn't really change because it is a counter. The kun and on yomi readings remain, but that is the same as for any word.
If you're stuck in a conversation while thinking of the correct counter, you can use the Japanese numbers (hitotsu, futatsu, mitsu, etc) to make yourself understood.
I think he means the slight pronunciation variations, such that you say "nihon" but "ippon" or "yon-mai" but "yo nin" and sometimes you must use the shi/shichi forms and sometimes yo(n)/nana and so on.
Your advice to learn the hitotsu ("wago") series as a default for objects (not people!) is a great one. I would add that you can use the "ko" counter as a default for numbers beyond 10 but I would still not insist on learning by heart even that default system as a priority.
Your point about numbers not being particularly important and even much less so than other vocabulary items (basic verbs, for instance) is absolutely core
In case it takes some time for Steve to reply: he has stated several times that he doesn't care about learning numbers in any language, at least in the beginning, because they're confusing and really not very helpful. In practice, if you go buy something in the country, chances are you won't understand [numbers are the most typical thing to be "jungle-pronounced", anyway*] and will have to rely on written tags or displays [learn to say "please show me the price" and "let me write the number" instead], no matter how hard you've studied them previously. That's my experience in several languages as well.. I would argue that not prioritizing numbers is even more important for a language such as Japanese: I feel that I'm becoming more confident in the use of counters and their various pronunciations over time, by learning them just as any other piece of vocabulary, i.e., through input. Among the many flaws of typical textbooks is their implicit assumption that mastering all the intricacies of numbers is among the first things you should achieve.
* About jungle, greenhouse and garden pronunciations, which is an important idea, IMO:
Very interesting explanation about the greenhouse vs garden vs jungle pronounciations. By the Russian Grammar channel! I used to watch that channel a lot some time ago, but I had never watched this episode... precisely because numbers are usually boring to learn, so I skipped it! So thanks for pointing it out