Almost any mobile hardware will do. Search for Wherigo and your desired OS (Android, Apple, WinMobile, Symbian, GARMIN, ...). You are looking for Wherigo-player software. Often the term WIG is used for "W"here "I" "G"o.
Android - Whereyougo
Symbian - OpenWIG
Apple - used to be named PiGo, now Apple Wherigo
Garmin - Garmin Wherigo player comes free with Colorado or Oregon.
If you have one, that's the way to go. Whereyougo will do the trick in that case.
Hard to tell you "the simplest method". Since there are so many options we need to know your existing hardware, your background and experience. If you have a Garmin (Oregon, Colorado) GPS, it is straight forward. Download a cartridge, copy it to the Wherigo folder on the GPS and start playing (Garmin does however seem to have stopped supporting the software, bug fixes are still pending).
Use whatever you have. If you have nothing I suggest to get a used Android smartphone with GPS plus Whereyougo player software. An alternative is a Nokia smartphone and the Symbian OS running OpenWIG. These solutions should cost you around 60$ if you buy used and smart. You might be able to pick up a used watertight Garmin Colorado or Oregon GPS.
That is hard to answer but I guess Garmin Colorado, Oregon with Garmin Wherigo software might be a likely candidate for pole position. The WIG player came with the Garmin device. Second rank is most likely Android and Whereyougo followed by Apple iPhones running PiGo (now Apple Wherigo)
Sure is.
Should be possible (legally unlocked, without contract), you might have to use a prepaid plan to make the phone work. Better ask someone who knows his mobile phones and contracts.