Connections
It's not wholly clear to me whether you are asking about booking and ticketing a connection or whether you are asking about how they actually work when you are at the airport.
Firstly buying the flights. It can be cheaper to buy the flight from A to B and the connecting flight (B to C) separately. Even though they could be in a single booking paying for them separately means that you get two tickets and two contracts. In the past many airlines would look at those two tickets, recognise the through journey and treat you as a through passenger. However, increasingly, airlines are treating each ticket separately. Why is this important? Because if you buy one ticket covering the whole journey the delivering carrier at the connecting point is responsible for finding you a flight to the destination should you miss the connection. Buy two tickets and miss the connection and the delivering carrier washes their hand and the onward carrier is likely to say "buy a new ticket".
Once you have your ticket the process starts when you check in for the first flight. Even when you are travelling on two separate airlines you will find that when you check in a tag is put on your bag that says "this bag is going from A to B to C". You will usually (but not always) also get boarding passes for both flights.
If you are changing planes in the UK on the way to the US when you get to your connection point you follow the signs for connecting flights, through security (again) and then to your new gate. There is the possibility that you might need to change terminals but the signs will lead you to the bus that runs between the terminals.
If you are doing it in the US you pass through immigration and collect your bag and go through customs. As you exit customs you will find check in desks and you give them your bag. Ask the people there where you have to go next. It's not easy to generalise but the likelihood is that you will need to find the transport to another terminal (or another part of the same terminal - they can be vast). You will go through security again and then to your gate. The real issue here is the time it takes to get through immigration. But the distance to the connecting flight can also be an issue. The airlines will sell 90 minute connections in some places. Personally, even allowing for the "it's all on one ticket so they'll find me an alternative" I would allow 150-180 minutes minimum at a US gateway. My record for plane to street was 15 minutes at San Francisco and I did it in 30 last October at Baltimore however, I've also taken up to 90 minutes just to get aircraft to street.
Coming home much the same applies except that when you change at a US airport you don't go through Customs or immigration. That said the US do want you to confirm you've exited the country. Generally this means making sure the airline takes back your entry permit but in some airports (Denver was one I saw it in) there is also a machine you have to visit *as well*.
In the UK you go through immigration when your plane from teh US lands but customs happens at your final destination usually.
For what it's worth, I tend to avoid connecting flights. I'm lucky that I've always lived within striking distance of Heathrow and Gatwick but even then I've tried to avoid connecting flights wherever I'm going. I'll tell you a story from 1978 (so it may no longer be relevant). I was going London/Frankfurt/Delhi/Kathmandu using Lufthansa to Delhi then Royal Nepal Airlines. Lufthansa in London tagged the bag to Kathmandu and gave me boarding passes for both their flights. I arrived in Delhi and found I had to enter India to make the connection. Got through immigration (took a while in those days!) and went to the RNAC check in desk. "Your bag?". I showed them my receipt, "Come with me" was the reply. I was taken past passport control, through to the departures lounge. "Can you see your bag?" and sure enough, there, against a pillar, was my bag. We took it back to check in, I got a new tag and new boarding pass and flew to Kathmandu where my bag arrived safely. Like I said that was 1978 and it wasn't the UK or the USA but it shows that (1) most airlines, like Lufthansa, do succeed in through checking baggage because the bag got through Frankfurt to Delhi no problem. (2) things can go awry (if not actually wrong) at connecting points.