Thank you so much! Is there a list of compatible phones for this deal? Where would I find that list?
See the link I posted above to the (very long) Slick Deals thread about the Sprint promotion. There is a list of phones on the first page.
I buy an unlocked phone here in the US. It can be any unlocked phone? Or on specifically compatible with Vodafone?
Any phone that accepts a SIM card should be compatible with Vodafone. In the US, we have two types of providers: CDMA (Sprint and Verizon) and GSM (T-Mobile and AT&T). In Europe, they are all GSM. (Technically, they are all LTE now too - but that's a more detailed discussion.) Any phone that is compatible with Vodafone will be compatible with every other European mobile service.
Older US phones for Sprint and Verizon were CDMA-only and might not work in Europe. Newer smart phones for Verizon and Sprint can also roam on GSM networks (like Vodafone, etc.). One possible "gotcha" is that Europe and the US use different mobile frequencies. A phone needs to work with the frequencies available where you are using it. This is less and less of a problem as newer phones support the radio frequencies used in both places.
Example: my old Moto E2 from two years ago for Verizon was a terrible phone for Europe - it would only work on the slow European mobile networks. It was a "fast" US phone. But two years, later, the Moto E4 has most of the "fast" European frequencies too.
You MIGHT find some Android phone (esp if a used or older phone) that might not work in Europe or not well. If you have any doubts...dig up the specs on the phone and verify. I can help decipher that for you. But if the phone you buy was just introduced this year or last, I can almost guarantee it will work in Europe. But...must be "unlocked."
I use wifi (what I do now with an old phone I have) to download the apps and such.
Yes.
Take it with me to Amsterdam. In AMS, buy a Vodafone SIM card for my phone from the USA.
Yes.
This sim card will give me service? Or do I still need a plan that can be purchased at AMS? Or does it just bill my credit card for everything used on the phone as it is used?
The SIM card gives you service. But, you'll be buying a prepaid SIM card. You buy credit ahead of time with your credit card. When the credit runs out, your phone will stop working. This is different from a "post-paid" service (What most phone users in the US have with Verizon, Sprint etc.) where you are billed at the end of each month for service.
Vodafone lets you use the SIM by charging you per minute, per text, per MB of data used...OR you can buy what's called a "bundle" with your credit. Vodafone for example has what they call "You" bundles that include a certain amount of data good for 30 days. gain - you buy that up front. In May when I was in Portugal, I bought a 2GB "You" bundle for my Dutch Vodafone SIM to give me 2GB of data for 30 days. (I was there only nine so that was way more than needed but the least I could buy -cost me 10 euros). The bundle doesn't give you minutes to call people but you can still make calls for 20 cents/minute, using whatever Vodafone credit you have left on the SIM.
You can add more credit ("top up") online on Vodafone's website (Dutch website only - it matters). I use Google Chrome which translates Dutch text automatically. I created an account for my Vodaone SIM and can add credit with my US credit card on that website, which applies it to my SIM card...so I can buy those bundles for 30 days or make European calls for 20 cents/min if need be.
Here is some more info about Dutch SIM cards including the Vodafone SIM:
http://prepaid-data-sim-card.wikia.com/wiki/Netherlands