Travel visas have turned out to be a bit more complicated than we thought. Our biggest challenge has been the timing of when to submit our applications for the countries we'll be needing visas for. For most countries, you cannot submit your visa application more than 90 days prior to entering the country. The problem with this is that we will already be abroad when we will need to send in our passports for visas. We were trying to figure out complicated plans like paying for same day service and use a courier via London. The timing was still an issue, we don't plan on being in London for more than a week and some counties like Kenya or Brazil for example can take a week to process a visa.
The solution is that you can apply for a second passport. We used a visa/passport expediter to help with this. We just received our second passports today, which is a relief. Now we will be able to send one passport in from wherever in the world while still having a passport in our possession. Basically we will be "leap frogging" our passports throughout our travels getting different visas we will need 90 days out. FedEx will be our friend.
These are definitely the countries we need to apply for in advance: Brazil, Kenya, Russia, China, India
There are many more that we would like to apply for now to minimize the headache of acquiring one in customs: Zambia, Tanzania, Egypt, Turkey, Nepal, Cambodia, Laos, Indonesia
The US department of State has a great website to explain all entry requirements to any country here.