Before the advent of computers, the only way of learning birdsong, I found, was to focus on a bird with your binoculars and listen to it! Over time you will become atuned to the familiar songs in the garden eg blackbird, robin, dunnock, blue tit, great tit etc. Then, when you hear something unfamiliar, get the binocs our, focus on your unknown sound, find the bird and listen! Obviously now, you can log on to the sites as listed below and learn your birdsongs a lot quicker! But be aware, there can be regional variations !! For instance, I used to live in NW England then when I moved to East Anglia I discovered that Great Tits, for example, had a slightly different song. Weird, I know, but it took some getting used to!
OK, it also helps if you have a "musical ear" (which I have!), but don't despair. With time and patience you will learn just about all the "cacophony" in your garden. I bet you'll be able to distinguish at least 2/3rds of your garden birdsong very quickly. I'm still learning new sounds and I've been a keen birdwatcher for 40+ years .... but then age is now against me, as I'm starting to forget....!!! lol.
Enjoy your birdsong - make use of the websites suggested and also try YouTube - it's amazing what you can find on there!
KK.