Best mobile and internet options when visiting UK

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?

At whatever airport you land at, there will almost certainly be at least one store (and probably many) selling SIM cards with included plans. You will be the best judge of what you are willing to pay for and thus which to purchase, but you can see example plans for Three, EE, and O2, which are some of the major mobile networks in the UK. All networks should provide decent coverage anywhere a tourist might go, so you might as well go for the cheapest option that meets your needs.

As you say you have an iPhone 5S, you'll need a SIM card that comes in the nano size. Just look for the word "nano", or for a statement of compatibility with modern iPhones (all iPhones from the iPhone 5 through the current iPhone 7 use a nano SIM). If in doubt, ask store staff for help.

To insert the new SIM card into your phone, look for the small pin-hole on the right-hand side of your phone. Use a paperclip to poke that hole; a tray will pop out, and you'll need to replace your existing AT&T SIM card with your new UK SIM card (don't worry about orientation, it only fits in one way, which is obvious). Make sure you hang onto your AT&T SIM, as you'll need to pop it back in once you're back in the US.

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?

These will count as international calls for both you and your travelling companions (note you are dialing an international number, and they are taking a call while abroad). However, if they call you, you will not have to pay anything extra (on the other hand, they will have to pay both for calling while abroad and for dialing an international number).

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.

While personal hotspots are available for purchase (though you may need to stop by one of the mobile network stores), they can be annoying to setup (although store staff will probably be able to assist) and expensive. Hopefully free Internet is available where you're staying (and certainly WiFi is available for free in Starbucks and McDonalds', among other places); if you're not expecting to use your laptop on-the-go, I wouldn't recommend buying a hotspot.

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