We have a pretty good idea about which countries we will potentially visit. We don't know exactly when we will be going to these places, but an important preliminary step is to research the tourist visa requirements for each of these countries.
Carrying a US and Japanese passport makes this process easy. Our passports provide automatic tourist entry into most countries without hassle - a privilege that some nationalities do not have.
After researching the requirements through websites such as www.visaHQ.com and going to each country's embassy or consulate websites, we think we only need to obtain visas in the following 4 countries, out of the 20+ we will be visiting. Not bad.
A note of caution: being too proactive by obtaining visas in advance will not help you. Many of them have an expiration date from the date of issue. If the date of issue is too soon, the visa will expire long before you actually arrive! Therefore, the application process will need to be done while en route. Many countries also require that there is at least 6-months left on your passport before it expires. I guess they don't want us squatting in their country without the possibility of being able to return home! The cost of the visas vary widely from country to country, but may dependon the passport you have. It's still good to research the process in advance so you know how much to anticipate and cost. Also, this is information we found currently, but the world is ever-changing. Best to look up the most up-to-date information as requirements can change.
Usual required documents for application include passport (of course), permanent visa card, passport photos (we plan to carry a set of photos for this purpose), round trip ticket to show we have an exit plan, proof of residency in home country via a utility bill, and bank statement (to show you have plenty of $ to support yourself and get home without being a financial burden), a local phone number (maybe use the hotel phone number), etc.
Brazil: there is an online application that is accessible. They claim it only takes 2 days to obtain the visa so this will be on our to-do list while in Buenos Aires. The visa must be used within 90-days from issue.
Thailand: Thailand allows a 30-day entry without a visa. However, it's a popular destination for extended stay due to its affordability and being very backpacker friendly. A stay past 30-days requires a visa that must be used within 3-months of issue. Therefore, we can think about visiting a Thai embassy or consulate while we are trekking through Eastern Europe.
Vietnam: this application seems like just a formality. An eVisa is obtained online. The email confirmation letter is submitted upon arrival and that's it. We'll take care of this process while we are in Thailand.
Cambodia: again, an eVisa is issued online. The 1-month visa that costs $25 is effective upon entry, so technically, we can apply for this at any time. However, what's the rush? We will probably obtain while in Thailand.
We plan on spending a few moments in an internet cafe along our route to take care of the above requirements. Seems like one of the easier parts about our travels!