Mexico is both GSM and CDMA. You can use your Sprint phone there - not sure how good coverage is, but it will work.
http://shop2.sprint.com/assets/pdfs/en/support/guides/services/worldwide/Worldwide_eflyer.pdf
For Taiwan you will need GSM.
Talk to Sprint and see what they say. They can give you a SIM for your US Sprint account number for their international roaming service. Call them up and tell them what you are doing and see what they suggest.
You have basically three choices. Roam on a US carrier (simple, but expensive). Buy an international SIM and use that (complex because you have a new foriegn number and expensive) or use VOIP either with WiFi or with 3G data on your foreign SIM.
If I am not paying for the calls (ie. company business), then I'll take my AT&T phone with me and roam on it internationally. This is the simplest, easiest and most expensive solution. You don't need to worry about people getting forwarded. If you are roaming internationally and you are in Moscow and someone calls your US number, it will ring on your phone in Moscow automagically... you just pay ~$1.25/min for this bit of magic (or >$2/min if you aren't on a international roaming plan). Even when I am on company business, I tend not to do this though... unless I have to. It's just so expensive. If I have an hour long telephone meeting to attend, I'm not paying $60+ just to call in even if my company will pay for it.
I travel a lot internationally and my preferred method is to get an unlocked 3G phone, buy a SIM card there with unlimited or a lot of data and call using a VOIP client (like Skype). It's by far the cheapest method for outgoing calls. For incoming calls, I can leave the phone in the SIP client app if I know to expect a call, otherwise I tell people to call my international number and I'll call them back on my VOIP client (I use Callcentric usually). With WiFi or really good 3G, SIP (or Skype) works really really well. In 3G mode, it eats up the battery, but it works great and it's super cheap.
If you don't need call access all the time, then use your hotel's WiFi or WiFi at a cafe. Then bring your Sprint phone, use it on WiFi with the radio in "airplane mode" and call on Skype or whatever VOIP client you trust more than Skype (my Skype experience has been hit or miss). You can make and take calls on a US number through WiFi and VOIP... but this only works if you are near WiFi when you want to make/take a call.
Like I say, I usually use VOIP and either WiFi or 3G and it usually works well in major cities and I've done this in a lot of countries, and then I pay $0.02/min. plus whatever the SIM cost me but I need to tell people my international number, or have the Callcentric app running and then it magically just rings on my cell phone with my US number and I pay US rates even though I'm on the other side of the world.
If you need to buy an unlocked GSM phone, I'd recommend Newegg or Amazon or Ebay... in that order. A lot of people on Ebay don't seem to understand the term "unlocked" though. And make sure that you buy one that has the GSM/UMTS bands for the countries you will be in. Your phone needs to be able to operate on the frequencies of the countries you will be in to work.