Questions on Airalo eSIM card and physical SIM card

I used to have an old sim card not smartphone phone to use in Italy where I purchased the SIM cards when I got there, but now I am going with using my actual smartphone while over there this time. I was looking at airalo, but I realized that they were data only and I couldn't make calls, and I have almost settled on being ok with using my AT&T plan.

I would go with a Europe wide company, like Vodafone as they MAY have better rates in other countries vs TIM (I used TIM when I lived in Italy and at least one time on another trip).

I would wait to get the other SIM cards when you get there as trying todo it from afar when you need to get a physical SIM card is just adding another step which is unnecessary.

No help to offer you on the other questions.

We will be visiting Italy and Denmark, and I plan to purchase the
Regional eSIM card (Europe). Once it's activated in Italy, do I need
to do anything else when arriving in Denmark?

A lot depends on the plan, but no, you should not have to do anything. You might get a welcome message when you travel to the next country, but with a regional plan, you should be fine.

Will all my apps on my phone still work when I'm on the eSIM?

Yes, your phone apps should work as they normally do. Apps will use whichever SIM or eSIM is active.

Will I still receive iMessage on the current number? I understand text is not included in the Airalo plan, do I need to worry about receiving text on my old number? Will I be charged for receiving text (sorry this might be a silly question). What settings do I need to change if I do not want to receive text when I'm in Europe (except for iMessages).

Apple Messages can be sent using data only, so when you have wifi, it's easy to send and receive them. This also might help:
https://www.airalo.com/help/troubleshooting/0UEL63PDK5IJ/can-i-use-imessage-with-my-esim/ML917DNVIN9J

I usually turn off all text notifications while I'm gone, and you can do that in your settings for any apps you receive notifications from (like banks, etc.). By the way, Apple (and Airalo) have good information on their websites if you get stuck.
https://support.apple.com/en-us/HT212780
https://www.airalo.com/help/

And this post might have some helpful info for you as well: https://community.ricksteves.com/travel-forum/tech-tips/newbie-to-esim-and-dual-sim-please-help

What settings do I need to change if I do not want to receive text when I'm in Europe

Under Cellular, after you install your esim, you will have a place to toggle your home plan to “Off”. That keeps texts and calls from coming through using your U.S. data plan.

Katharine, just to let you know, Airalo does have a global esim that includes phone calls, in case your phone doesn’t have a physical SIM card slot. It is more expensive, of course, but might be an option. I generally use a data only esim, and Messenger or Skype if I need to make calls. Rarely I expect to need to receive calls from European numbers while in Europe in places WhatsApp is not prevalent. So I have used the global esim once.

Thank you everyone for the detailed replies. Really appreciate it!

@ Southmelon

I am not an iPhone user. At present I am in Puglia near Martina Franca Italy, and this is my experience with my mobile phone.
I have a dual sim Nokia. On 19th April, in Venice, I purchased a TIM Tourist pass for 30 days to cover my time here. It cost 30 Euros.

This is the message from TIM:

Tourist Pass Full active: you have 100GB internet in Italy, 15GB in Roaming in UE, unlimited minutes in Italy and 100 minutes to call abroad for 30 Days. Only once you will be in Switzerland you will have 3 GIGA for 30 Days.

This is a direct copy even though the grammar is not Oxford style.

My Australian SIM is set to receive SMS only as I need to receive numeric codes to verify my identification (3FA and 4FA) for financial transactions. All other, data calls etc are set for the Italian TIM SIM. Should be a Youtube video on how to do this for iPhone. I access free WIFI when available and appropriate. Important contacts in Australia have been advised via email of my TIM number for voice communications.

When I first started to need O/S SIMs and Dual SIMs, my Australian Telstra mobile phone account manager helped set things up and led me by the hand through the process, trips and traps.

I have chosen not to use ESIMs yet.

Have not tried to use an Italian TIM SIM in Denmark so do not know if they transpose.

Hope this helps a little.

Regards Ron

Thank you @Ron!