Some countries require visas prior to arrival, some do not, and visas expire at different times. The US State Department has a good summary of travel requirements for US citizens. I started with this list, then visited each of the countries' embassy websites to confirm the requirements and compiled this into a table. The important countries are those that do not grant a visa on arrival. Given my itinerary, these are Australia, China, India, Tanzania, and Vietnam. Australia just requires an electronic authorization (ETA). China has more extensive requirements, but according to the Lonely Planet forums, it is not hard to get a visa. Tanzania is so far out on my route that the visa would expire before arrival, so I will need to get it while on the road. I will either get the visa for Vietnam prior to departure or, like Tanzania, from one of the embassies on the road. I plan to get the visas for China and India during the last month before departure from the US.
For Chinese visas, you have to apply in person (no mail-in service). This is a hastle, who has time to drag into Washington or NYC? Well, I might have time to do this in person in NY in August. A LP post recommends China Visa Service Center. Fees start at ~$29, then add mail to/from for another ~$30 on top of the embassy fees, so we're talking an excess of ~$50 for a few hours in NY. The consulate for the north east is in NYC at 520 12th Ave., New York,NY 10036 212-244 9456. A 6mo double entry visa is $75 or multiple entry $100. From the embassy site: "The regular processing time is 4 working days. For express service, additional fees of $30 for 1 working day processing (applications presented before 11:30am may be picked up between 2:30pm-3:00pm on the same day), or $20 for 2-3 working days processing will be charged." So really, $50-$30 = $20, which is still worth a few hours hanging out in camera shops or sightseeing in NYC.
For India, the application asks for references, but apparently this does not apply to simple tourist visas, see this LP post. The NY consulate handles the North East, at 3 East, 64th Street, between Madison and Fifth Avenue, New York, NY10021, while the South East is handled by the DC consulate at 2536, Massachusetts Avenue, N.W., Washington D.C. 20008 (202) 939-9806. For the embassy website: "Applications received at the counter between 9.30 a.m. and 12.30 p.m. are normally processed the same day and serviced documents can be collected from the counter between 4.30 p.m. and 5.30 p.m. Applications sent in mail will normally take about 5 working/business days to process from the date of receipt in the Embassy." Since I will be stopping through NYC later in August I can just do this in person and save time and mailing fees.
Indian consulate in NY
Chinese consulate in NY
For Australia I need an ETA ~$20, good for visits up to 3 months within 12 months of issue, multiple entry. Application is online.