I will probably buy an unlocked phone and bring it with me. Do I just buy a sim card at any phone stores? Will that come with unlimited data? And can that be used in different European countries?
Yes, you can just buy a prepaid SIM card at a store in the first country you visit. No, it probably won't have unlimited data. But unless you are streaming video and/or none of your lodgings have WiFi. Last year, I did a 17 day trip in Europe and bought a Dutch Vodafone SIM ahead of time on eBay and bought a "You" bundle to give me 3GB of data for 30 days. In 17 days, I used only 2GB.
Mobile plans vary by the company and the country. I would make sure you can use your data with hotspot and that they allow roaming in Europe. EU rules state that mobile companies can't charge extra for roaming within the EU outside of the home country, but that doesn't mean they have to offer roaming - just can't charge more for it. Just ask at the mobile store when you buy it - they'll set it all up for you before you leave the store. If you have a tablet or laptop, take it to the store with you, test out your hotspot on the spot and make sure it works.
Based on my experience with the Dutch Vodafone SIM, I can recommend it (I used hotspot a lot last year), especially if you happen to be visiting the Netherlands first on your trip. You could even buy it on eBay ahead of time like did, but you may not need to if you will visit there first.
Are you certain your Verizon phone is locked? Verizon (unlike AT&T) historically has not locked its postpaid phones (that is changing very soon I believe per Verizon's announcement but that shouldn't affect older phones). Verizon prepaid phones are locked, though. If you buy a phone in the US, make sure it has LTE frequencies for Europe - not all of them do. You could buy a phone when you get to Europe. But do confirm that your Verizon phone really isn't unlocked first.