Here is one from a tourist's perspective.

Everyone tells me to "just get a sim card" when I arrive in the UK. Never having had an unlocked phone before, I wouldn't know where to begin. I've never even seen a sim card. How does one go about selecting a pay-as-you-go mobile plan that includes calling/text/data?

Go to any decent marke and you will find a number of shops selling SIM cards. You can request them to initialize the new settings for you and generally they'd be happy to do. What package you need is subjective, I personally bought an EE - suggested by the seller, 1 week ago in London and it worked Great for me, I mostly needed it for Data.

Assuming I sort this out, can I use my UK phone number to make local calls to my traveling companions who will also have US phones (yet may not be equipped with UK sim cards)? Or would that qualify as an international call, even if the person is located on the other side of London?

The calls to US numbers will be international, how much you get charged for those depends upon the carrier and the package - don't forget to ask them about it before buying.

Then there is the matter of internet options for both mobile and laptop. Some sort of personal hotspot? Whatever it is, it can't be something that I need to struggle with for days just figuring out how to use and troubleshoot. It needs to be something that even a child can manage. I want to enjoy my trip, not spend most of it doing my own personal tech support gig.

Recent iPhones have personal Hotspot option, you just turn the hotspot on and search for it on your laptop and authenticate. I was doing exactly that for a whole week.

Another great thing is that EE provided me with WiFi on the tube as well because I was using their connection. Generally on the underground you don't always have signals for your mobile data won't work and that WiFi is a blessing. I am sure other carriers also provide such a service but my experience was with EE only.

Although as others have pointed out there are plenty of SIM dispensers at the airports but since you have many questions in mind just like I did, I would suggest you buy the SIM from a shop where a person attends you. This method has worked fine for me in many countries

Here is one from a tourist's perspective.

Everyone tells me to "just get a sim card" when I arrive in the UK. Never having had an unlocked phone before, I wouldn't know where to begin. I've never even seen a sim card. How does one go about selecting a pay-as-you-go mobile plan that includes calling/text/data?

Go to any decent market and you will find a number of shops selling SIM cards. You can request them to initialize the new settings for you and generally they'd be happy to do so. What package you need is subjective, I personally bought an EE - suggested by the seller, 1 week ago in London and it worked Great for me, I mostly needed it for Data.

Assuming I sort this out, can I use my UK phone number to make local calls to my traveling companions who will also have US phones (yet may not be equipped with UK sim cards)? Or would that qualify as an international call, even if the person is located on the other side of London?

The calls to US numbers will be international, how much you get charged for those depends upon the carrier and the package - don't forget to ask them about it before buying.

Then there is the matter of internet options for both mobile and laptop. Some sort of personal hotspot? Whatever it is, it can't be something that I need to struggle with for days just figuring out how to use and troubleshoot. It needs to be something that even a child can manage. I want to enjoy my trip, not spend most of it doing my own personal tech support gig.

Recent iPhones have personal Hotspot option, you just turn the hotspot on and search for it on your laptop and authenticate. I was doing exactly that for a whole week.

Another great thing was that EE provided me with WiFi on the tube as well because I was using their connection. Generally on the underground you don't always have signals so your mobile data won't work and that WiFi is a blessing if you're bored. I am sure other carriers also provide such a service but my experience was with EE only.

Although as others have pointed out there are plenty of SIM dispensers at the airports but since you have many questions in mind just like I did, I would suggest you buy the SIM from a shop where a person attends you. This method has worked fine for me in many countries