We will be travelling internationally but want to avoid roaming charges from our current carrier. A couple of possible solutions: 1. Keep our phones on “Airplane Mode”. Can we receive SMS messages in this mode?

Kabby-1 wrote:

To avoid roaming charges when we travel internationally, can we leave our phones on “Airplane Mode”?

Sort of. While in airplane mode, no roaming. If the iPhone restarts, cellular will be active, and you’ll get roaming charges.

If so, will we still receive SMS messages?

No. SMS is a cellular function and requires carrier access. Which means roaming.

Apple Messages app, and other such can arrive via Wi-Fi, when that is available.

Some carriers might forward your messages via Wi-Fi when they have support for Wi-Fi assist, but I’d not assume that availability without checking with the carrier.

Consider getting an eSIM from a local carrier in the country or region where you are traveling, if those eSIMs are available, or from a carrier that has international roaming in place in the intended country or region. Some regions can have better local roaming plans and pricing than others too. Across Europe with a local carrier eSIM roaming for instance. The US has comparatively poor and extractive coverage, if you’re traveling here.

Not until you take the phone off of Airplane Mode. Even then, you won't get the message unless the phone is connected to the Internet.

So, if the phone cannot connect to local cell network (it won't unless you have have international roaming) you will have to find a Wi-Fi Hotspot.