Around the world trip, is there a way to simplify the visa gathering?

The keys to solving visa problems on such long trips are (a) planning/research and (b) getting visas on the road, as you travel. Getting visas to "all the countries in the world" in one shot and then embarking on a trip is just not realistic.

• First, take a look at http://en.wikipedia.org/wiki/Visa_requirements_for_Belgian_citizens
• Note that for Belgian citizens, only a handful of countries require visas. The rest either don't require a visa, or have a visa-on-arrival process, or require some sort of easy-to-get electronic permit which you can apply for online, etc.
• Pay close attention to the fine print for these "visa-free" and "visa-on-arrival" countries. For example, do you have to arrive via a particular entry point (typically the main airport in the capital city) to get the VOA? Do you have to have onwards tickets? What's the maximum stay? Aside from the Wikipedia link above, this is a good source of information if you're travelling by air: http://www.iatatravelcentre.com/
• For any countries that do require visas, find out where you are allowed to apply for them. Some countries are easy (e.g. you can get a Chinese visa pretty much anywhere in the world as long as you meet the requirements, like onwards tickets), while some countries may actually require you to apply in your home country. Plan your trip accordingly.
• Sometimes, even if you can't get a particular visa in a foreign country, you may be able to mail your passport back to the country's consulate in Belgium for a visa (or better yet, to a visa agent in Belgium which will handle the consulate interaction for you). Of course, mailing away your passport while in a foreign country is never fun (or cheap), so use this only if absolutely necessary.
• Keep timing in mind. Some visas you can only get between X and Y days before your intended trip, so you must time your visa application correctly.
• Keep in mind that some visa application processes elsewhere in the world may not be what you're used to. Be ready for crazy/inconsistent requirements, long processing times (processing time = "whenever the consular officer feels like taking a look at your file"), even bribe requests, etc. I've had some pretty unusual visa application experiences and have heard of even crazier ones. Read about people's experiences online, and if a given visa process is particularly crazy, apply for that visa in Belgium since the process in your home country will likely be the easiest and most efficient one.
• Keep passport space in mind. You may need to go through several passports to have enough space for all the visas and stamps, and many countries require 2 blank pages before they'll issue a visa. I don't know the rules in Belgium - you may need to go back in the middle of the trip to change your passport. Don't be shy telling immigration officers around the world to conserve passport space (tell them which page you'd like the stamp on).
• There are lots of other nuances as well, e.g. the ability to enter countries more than once in a short period of time (e.g. this can be an issue with India), requirements to have a booked tour (Bhutan), the need for an official "invitation" which can typically be arranged through a 3rd-party company (e.g. ex-Soviet countries), etc.

Basically, either do lots of detailed planning (if you have a fixed amount of time and know in advance what you'd like to see), or just take a more relaxed approach, learning as you go, getting visas as you go, but knowing that you may run into unexpected problems and may not hit every country you may want. Personally, having tried both approaches, I think the latter approach is more fun for super-long trips, and you can't plan for everything anyway.