It is an old topic, but it still relevant question. First time I bough NEM using VISA card with USD from changelly. Although is not the cheapest way to do it, but for sure easiest. If you want to save some money I would suggest to buy xem by using XEM/ETH or XEM/BTC pair.
I did a comparison of 4 most popular exchange platforms which offer XEM/NEM trade: Changelly, Zaif, Binance, Poloniex. Only Changelly accepts fiat currencies, but as I mentioned earlier, it is not the cheapest way. So you could take a look at the article -
https://get-cryptocurrency.net/where-to-buy-xem/ and decide which exchange platform suits you best.
The main criteria for comparison: payment options, transaction fee, allowed/restricted countries, registration/ approval of account, fiat currencies accepted, limits. Shall I add some more criteria for comparison?
I hope this summary will help you.
Please let me know if you see some wrong information or if am I missing something. Also let me know if should add some other exchange platform to this article.