The Nintendo Switch was announced, among other things, as a Region-Free system, which would let the users to purchase content from any region in the world, and play it on their Switch. This is true, and it works with no issues whatsoever when using physical copies of the games, but it's a bit tricky to get it up and running when using digital content.
First, let's see how Nintendo treats their world regions for game purchasing in the eShop:
Americas: United States, Canada and Mexico (they left the remaining Latin America region out). They will add Argentina in the near future as well.
Europe: Europe, South Africa, Australia and maybe others (some people consider Australia as an entire different region, it doesn't matter at all).
Japan: Self explainatory
These are the three main regions from you can buy content on your Switch. Depending on your country you may or may not have access to an eShop, and online servers, but given this region-free nature of the console, it can be easily bypassed.
The simplest way to buy content from another region (and use servers from other countries) is to change your Nintendo Account region.
In case you didn't know, the digital games are now tied to your Nintendo Account, not to your individual system as it was with previous Nintendo consoles. This allows a player to purchase a new system, and download their content to it, disabling the previous one first, of course, as Nintendo will only let you download and use the games you download on your "main" console.
Switching your Region
There are two methods of purchasing content from another region on your Nintendo Switch, both which I'll explain here.
Single Account
This is the least common method I've seen in use, but in my opinion is the best, since you have all your purchases on a single account, you need only one single profile on your Switch, and your My Nintendo rewards points will be tied to just a single account. There's just an issue with this method: You cannot switch regions if you have any money in your account, you must find a way to empty your balance before switching because Nintendo won't let you do it.
Go to accounts.nintendo.com/ and login if needed
Enter the Profile Section
There you can change your Country/Region (the timezone will shift accordingly).
Save your Changes
Now, go to your Switch and open the eShop. It will ask you to login again, you should do so. Depending on the selected region it may or may not be able to display content with your system's settings, and it will tell you it will change to a compatible setting (perhaps your Switch is in english and you're trying to view the japanese eShop).
To switch eShops you need to change your profile region and login again every time.
Multiple Accounts
This is the most widespread method, the easiest, but the clunkiest since you'll have more than one account on your Switch, the games will be spread among those accounts (remember your Switch will only let you create 8 profiles), and your My Nintendo Rewards points won't be consolidated on a single account. This is the recommended method if there's no way to empty your account balance.
You need to create a new Nintendo Account from a computer or a smartphone by just going to nintendo.com and select the Signup button. After you create your new account with a different region, go to your Switch and create a new profile, and Login with that profile you just created.
To switch eShops you'll just need to select the region profile you want and you'll be shown the selected eShop.
Purchasing Content
So, you can now browse the different eShops, but you want to download something more than just the demos. There are two main methods to do this: Purchase eShop currency with either eShop cards or standard credit cards (credit cards go in here because you're basically purchasing eShop currency to pay for your game), and Purchase Download Codes.
Credit Card
If you have an international VISA/MasterCard, you can try to purchase your content directly from the eShop, and depending on the region restrictions you may or may not be able to complete the purchase (a Mexican MasterCard can purchase, for example, in Mexico and USA, but not in Japan). Tip, if you're using the US eShop you may as well select an Alaska ZIP code to avoid paying taxes.
eShop currency
If you can't use a credit card on the eShop, you can go to Amazon and purchase "Nintendo Eshop Cards", depending on the country you can get cards on different currencies:
amazon.co.jp issues Yen eShop cards
amazon.com issues US Dollar eShop cards
amazon.co.uk issues British Pound eShop cards
amazon.fr issues Euro eShop cards
Remember that the eShop card option works best for the multiple account method I mentioned earlier, since there's a high chance of having residual balance after a purchase, and thus you wouldn't be able to switch your account to a different region.
Downloadable Codes
You can purchase downloadable codes from the same price as in the eShop (or less, as is the case of Japan) on Amazon and perhaps other vendors.
Amazon Japan constantly offers the Download Codes for Switch games with a 10% or higher discount from the retail price, while Amazon US sells them at the standard price (not Prime discount for you on downloadable games). The only caveat for Amazon purchases is that you need to have a (fake if you like) shipping address on the country of your choice. There are several websites to generate this information. Just make sure you're buying a download code and not a physical copy of the game or it will be sent to your fake address!
After you purchase your download code, Amazon will send it to your email, and display it on screen. Just go to the eShop from the region you just purchased your code (this is very important), and select the redeem option. After a quick check, you'll be able to download your content.
Caveats and Answers
One thing I didn't mention is that the moment you activate a new region account on your Switch you'll start receiving news from that region. It won't hurt, but they'll be there and they'll be mixed with your regular account news (perhaps in japanese or other language), just deal with it.
Another thing that's quite often asked is that you can purchase DLC content from a region and apply it to another region game. The greatest example is Breath of the Wild, you can get the US version of the game and go to Amazon Japan to purchase the DLC code, and it will work.
To help you with the purchase of digital content in other countries Amazon Pages, you can use Google Translate Chrome extension, or the smartphone app. In the case of Japan, there's a small world icon in the middle of the menu which can be used to display menus and other information in english to help you.
Last, but not least, you need to be informed of what you're purchasing. The japanese games bought via Download Code on Amazon are the cheapest ones most of the time, but you need to be aware that not all games support every language. You can check the language support on the eShop page of the game you're trying to buy, but you can also visit this page: http://www.perfectly-nintendo.com/nintendo-switch-list-of-games-with-english-language-option-in-japan/ It has a comprehensive list of language support in Switch games and will help you in your hunting.
All this information has been researched by me personally, I use the one account method, and even got the japanese Zelda DLC. One of the DLC for Blaster Master Zero was bought in Mexican pesos and the other in Yen. I also got Kamiko using a 500 Yen Japanese eShop purchased from Amazon Japan, so I can tell you that this works with no issues at all.