We're planning on bringing our dog with us for a year in Europe. Although we'll settle in 3 or 4 countries, we'll travel on weekends. We've started the procedures for the England requirements (microchip, bloodwork, etc.). However, we will be spending time in Italy and Switzerland before hitting England.

Do we get the official "pet passport" while in Europe? Or is having the required paperwork documenting vaccinations enough?

Having the required paperwork should be enough. You will need to check with the individual countries on their requirements, with particular reference to the country you will be entering from the US. But you may want to get the pet passport after you arrive, simply because it would make it easier for you to travel within Europe. Bear in mind also that Switzerland is not a member of the EU, so may not accept the EU pet passport.

There are additional requirements for getting your dog into the UK, beyond the pet passport. Pauline has written up an excellent step-by-step procedure for this, posted here, with links to the relevant DEFRA pages. Two things to note within this process are:
(1) The microchip must be a European-readable microchip (the US microchip format can be different from the standard EU format, as with so many electronic things). If the microchip is not the EU standard, then you'll need to purchase a scanner and carry it with you so the dog can be successfully scanned. Which might not be a bad idea anyway, because if the dog isn't "scannable," then it's automatically disqualified.
(2) The tick and tapeworm treatment must be done with specific approved drugs, in a very narrow time window. See the DEFRA site (included in Pauline's links) for specifics.

Good luck with all your travel plans and paperwork.

Thanks Kim! Will a country issue a EU pet passport to us even if we are from the U.S.? This would make it so much easier, but I'm not sure if they have restrictions since we are not from the EU.

Houseblend, I'm afraid I can't answer for the rules in each country. In England, we got the pet passport from our local vet. Of course, we were living there at the time, so had a UK address and a regular vet. My guess would be that you can do the same, if you will have a fixed address for some amount of time.

I'd suggest that you ask for contact info for a good local vet, preferably one who speaks English, as you make arrangements for your lodgings. (You'll want that info in case of emergencies anyway.) You could then contact the vet and ask what the procedure would be. Have you discussed the EU pet passport with your regular vet at home? Your vet might already have the info, or have contacts to simplify getting it.

We didn't use a pet import service ourselves, but a number of Americans used the Air Pets service to bring their dogs/cats into the UK from the US. There are other pet importers in the UK and other countries, and of course you can always google to find other companies/information. This US Embassy site has some information on importing pets into Italy, with other links.

We successfully took our dog from the US to Germany, then to the UK, and most recently to Italy, so I know it can be done. You're wise to have started the research process early, and as long as you have all your paperwork, you should be fine. Good luck with all your plans!