Zomp, I wouldn't get a MiFi to be honest. My wife and I took a trip in September to France and Spain and we just did the SIM card. MiFi's are notoriously unreliable. I've heard horror stories about how they don't work or have spotty coverage or have horrible battery life. Also, it means you have to carry around another device. It's just... yuck. I considered getting one, and after speaking with a buddy who travels internationally all the time, he convinced me that the SIM was 100% the way to go.
You definitely have access to your numbers and messages. Those are stored on your phone locally -- I was iMessaging and FaceTiming all over the place.
I purchased a SIM card at the airport in Barcelona -- I believe for 7GB of data + 60 min of voice it cost me 20 euros. And in France, it was similar -- I think I got 1GB of data and 30 min of talk (and I topped it off with 5 more GB of data) for about 25 euros. Both allowed me to top off my phone if I wanted, but I only had to in France. Heck, if you were going to Spain I'd send you the SIM from there -- I still have at least 30 min and 2 or 3 GB of data on it.
I believe that all Verizon phones come unlocked. And I believe that they should work just fine in Europe with a local SIM. But, if your phone isn't unlocked this is what I would do (and what I did with my AT&T phone): I called up my provider and explained to them that I've been a customer for 10+ years with them and that I'd like my phone to be unlocked early. I want it to be unlocked so that I can purchase a SIM overseas and use it to stay in touch with my wife in case we get separated while traveling. When they inevitably ask you why you don't want to use their overseas service, simply tell them that it's too expensive to consider using, and cite that for 1 GB of data over there you'd have to spend $20,000 -- and for that you could just buy three new unlocked phones. I had success talking to someone online rather than on the phone to accomplish this, btw. I had to chat with a few people and it took a couple of tries, but eventually someone was kind enough to unlock my phone.
BTW get yourself a safety pin and keep it pinned to your bag -- you can use it to pop open the sim tray very easily.
Edit: just re-read your post. Yeah, if you have a 5c I would just do the SIM. It takes 10-15 minutes in each country to deal with setting up an account w/ the provider and buying a SIM. Well worth the hassle of not using a MiFi.