No question you'll want an unlocked phone. "Unlocked" means not locked into any particular carrier's service; in other words, a phone locked to T-Mobile service generally won't work on Verizon. You'll want to make sure the phone, even unlocked, operates on the reception bands in use in the area where you'll travel. Most modern phones will, but you don't want to buy a phone and then get to your destination just to find out your particular phone won't work too well. This website will give you some idea of the parameters of that issue: https://www.gsmarena.com/.

Amazon is the best source for these phones since they carry a wide variety of new and used Androids/used iPhones and some models that aren't always designed for the American market but that will work very well in Europe and Asia. Keep in mind that if you go the Amazon route and you're a Prime member, there are rumors they'll do another Prime Day sometime in late June. At that time, they run some very attractive offers on many cell phones--you can probably get something that would meet your needs for less than $175.

Regarding the SIM to use, you'll have to do a bit of comparison research, keeping in mind where you are going and how well certain SIMs match up to the areas where you intend to travel. I do a lot of travel in the UK and Ireland and have settled on http://giffgaff.com for my needs, partially because in addition to coverage for my intended travel, they also accept American credit cards for payment.

This company and many others all over Europe run a pay-as-you-go service--in other words, no monthly/annual contract commitments. You buy a fixed period of service with a fixed amount of cell minutes and data. That setup is perfect for a one-off trip lasting several weeks although because of the frequency of my travels to the UK/Ireland, I've kept my service with giffgaff alive for several years now, taking the occasional step to keep my account going. As a result, most of the people who need to talk to me now have a phone number they know to use when I'm overseas. Giffgaff also works in the EU, but I suspect that setup works better if you use it mostly in the UK/Ireland with a few trips to the Continent rather than using it mostly or all of the time on the Continent.