Here is a procedure that you could follow, or modify to change the results. First you need a list of major national languages, which means figuring out what counts as being "national" and what is "major". In some countries there is an "official" language, or a "national" language, but the US doesn't have such a legal designation. People "intuitively" know that the national language of the US is English, because it's the language spoken by most people. The official national languages of Switzerland are German, French, Italian, and Romansh – but AFAIK, most people in Switzerland don't speak Romansch. The other problem is that there are small countries like Iceland and Estonia which have national languages, but which rank low in population statistics.
The second thing to do is aggregate similar languages. For example, Bengali and Hindi are different languages and they are both major national languages, and, they are similar enough that knowledge of one substantially carries over to the other – you could merge the two. Following that procedure, the 25 largest languages that meet the criteria are (in approximately population-descending order) are Chinese, Spanish, English, Arabic, Hindi, Portuguese, Russian, Japanese, Vietnamese, Turkish, Korean, French, German, Polish, Italian, Romanian, Azerbaijani, Farsi, Burmese, Serbo-Croatian, Thai, Dutch, Uzbek, Indonesian, Filipino.
You might knock a few languages off the list as somewhat redundant, for example you could limit the list to one Slavic language, one Turkic language, one Malayo-Polynesian language, one Romance language and one Germanic language, bringing the list down to 14 (Chinese, Spanish, English, Arabic, Hindi, Russian, Japanese, Vietnamese, Turkish, Korean, Farsi, Burmese, Thai, Indonesian).
An alternative approach is to lighten up on the criterial languages – consider all languages to be equally valid – and also reduce the acquisition standard to "elementary pronunciation competence". You don't have to become fluent in Japanese to be able to handle basic Japanese pronunciation. Also, "Arabic" is not just one language, there are many dialects, and the pronunciation of certain consonants such as [ʕ] differs substantially across dialects, thus you will not acquire a native pronunciation of Iraqi Arabic if you study Moroccan Arabic pronunciation. Likewise, the vowel æ in Norwegian and Swedish is pronounced differently, though the two languages are very similar (also, not phonetically very much like English).
An alternative is to train in the production and perception of the IPA, which is designed to at least enable you to make and perceive any language sound. This doesn't help with Georgian and Slavic consonant clusters. I suggest, then, a list of two: learn IPA, and learn to pronounce Georgian (which is a national language).