You need to first identify the country in which you want to reside, and then learn its immigration laws. The IATA site referenced by Andra is not a good source of information, it may have enough for tourists and business travelers, but that's not what you're going to be. I've just checked it for a couple of combinations that I can relate to and it didn't provide any useful results.

You would also need to deal with taxes, and plan it accordingly. There are tax treaties with certain countries, but not with others, that reduce your tax burden there, and in the US. If there are no tax treaties, you will be subject to double taxation. The US treaties won't reduce your US taxes, but might help you reducing or eliminating your local taxes in that country.

You should not compare yourself to volunteering organizations such as the Peace Corps, as they work in an entirely different legal framework. What their members/volunteers can do - you are not necessarily able to (and most likely - cannot, in most places).

Specifically to your questions:

What type of visas would be best suited for this type of extended tourist type travel?

This is not tourist type travel. You need a temp. resident/foreign worker visa, in most countries.

Where can I find information about which countries offer extended tourist/visitor visas?

Embassies.

Might it be feasible for me to set up a US LLC to obtain a Business visa to establish my business in a foreign country? If so, what sort of cost range would I expect for this type of visa?

Depends on the country, and you should consult an immigration attorney in that country. In your own country (the US) - this is not an option readily available for foreigners. Why would it be different elsewhere? My default assumption is that it wouldn't.

Does this plan sound feasible (from a visa perspective)? I don't hear of many people doing this, but that could be for a lot of reasons. Are there others that you know of who pull off something similar?

Doesn't sound feasible to me.