I had never used buses until 5 years ago when I moved to Scotland, so it was a new experience for me, too. No need to feel embarrassed, as it's a new situation, so you're understandably nervous. Unfortunately, I can really only tell you about Lothian Buses in Edinburgh, as I've only used First Buses a few times, but the buses tend to work the same, so this should help with Livingston as well.
Lothian Buses in Edinburgh are very easy to use, and we have one of Europe's best bus systems. Once you get the hang of it, you'll find using the bus to be very easy and convenient. You can plan your journey at http://travelinescotland.com (buses throughout Scotland or journeys from Lothian to Edinburgh) or http://lothianbuses.com (buses in the Edinburgh area). First Buses operate in Livingston, so you can see their timetables at firstgroup.com. There are also maps and timetables at the bus stop to assist you, but planning your journey beforehand online will be the best thing to do. Other people waiting at the bus stop should also be able to help you if you need help, as they probably travel by bus frequently.
In Edinburgh, the bus is £1.30 for a single journey and £3.20 for an all-day bus pass. So, if you're going to be on more than 2 buses, you should ask for the day ticket. You'll need exact change in Edinburgh. On First Buses in Livingston, the fare varies by how far you're going, so there isn't a set fare. When I've taken First Buses before, they do give change, so you won't need to have exact change.
Once you figure out where you're going and arrive at the bus stop, you'll see that some stops in Edinburgh have bus trackers that tell you when your bus will arrive. Some stops don't have this, but you'll find a timetable on the bus stop, which will tell you when your bus arrives. People tend to queue at the bus stop, so being off to the side is fine, if you're in a wheelchair.
When your bus is approaching, it will indicate that it's stopping by putting on its turn signal, or if it doesn't do this, you should flag it down to let the driver know you want to get on. Make sure you let all passengers get off the bus before you get on, and let people who were queuing ahead of you get on first. People will probably let you on first if you're in a wheelchair. The driver will lower the bus to let you on.
Once you're on the bus, you'll see that there's a window with a red box, where you put your coins. Simply put either £1.30 or £3.20 into the red box, and tell the driver that you'd like either a single ticket or a day ticket. The driver will enter your selection and a ticket will print out. Some buses will have a ticket print right next to the red box, whilst others will print out just behind the driver. If you're on a First Bus in Livingston, you'll tell the driver where you're going and he'll tell you the fare. Hand him your money, and he'll give you change. The drivers tend to be very helpful, so don't worry about not knowing what you're doing. They're used to tourists or people who haven't ridden the bus before. Most will be more than happy to let you know when you should get off, if you ask them.
Hold onto your ticket. If you have a single ticket, it's not really a big deal, unless an official from Lothian buses comes onto the bus and begins asking to see tickets, but this is rare. If you have a day ticket, hold onto it, as you'll need to show that to the driver when you get onto other buses.
Once you're on the bus, head to the pram/wheelchair area, which is clearly marked. If there's already a pram there, they should be able to move, as prams are smaller than wheelchairs and can fit in another place.
You should have an idea of where you need to stop, but if you miss your stop, it shouldn't be a problem, as bus stops aren't too far apart, so you can always get off at the next stop and go back to where you should have gotten off. Once you know that you'll need to get off, press the red "STOP" button. You should press the button after the bus has stopped and let on passengers at the stop before yours, and before the bus arrives at your stop.
To answer your question, if someone else presses the button first, you don't need to press the button again. Just make your way to the exit and exit with them. It's probably a good idea to let the driver know beforehand where you'll be getting off, as he will need to make sure you're able to get off with the wheelchair mechanism that allows you to exit (not sure what it's called).
Hope that helps!