Transport for London's goal with trains and buses is generally run them at least every 15 minutes at all times. Because as soon as you do, people stop looking at timetables and just turn up and go. I lived in south west London for many, many years. I never looked at the bus timetable. As long as it wasn't past midnight, I knew I could just turn up at the bus stop and get a bus within five minutes. Ten minutes max. Some of the daytime routes would run until 1am. New Years Eve? Everything ran through the night!
This is also why the Metrolink does so well. You don't need to think - you just get to a stop and there will be a tram within 12 minutes, max. It's turn up and go. It starts early, it runs late. It has a decent New Years Eve service. Compare that with Rose Hill's train service in the morning peak where someone's basically decided what time to run the trains by stabbing a pin in a piece of paper at random, and there's virtually nothing after 6:30
And as for fares...
London's fares aren't that cheap, but they're integrated. The same ticket covers trams, tubes, trains and buses. You buy your weeks season ticket for the tube, and you essentially get bus travel for free for the week. So you become more inclined to jump on that mega frequent bus instead of travelling by car. Here in Manchester you can buy a monthly train&bus ticket but its more expensive. And it doesn't cover the Metrolink for no good reason.
But it's not just frequency and fares. It's destination too. We do actually have a reasonable daytime bus service to Stockport. Five buses an hour is pretty good going. But unless you're going to Stockport (or Romiley) it's pretty poor. The evening services don't run as often. Rose Hill's evening service is a joke. My parents don't live more than a few miles away, but it would be a nightmare to travel to them by public transport despite them living on a bus route and a railway line, because nothing connects well. It would take three buses (good luck trying to get them to connect) or a train followed by a walk then a bus (ha!).
Basically a lot needs to change before we'll get people using public transport.
You need to replace all the season-tickets with one multi-modal ticket that covers everything.
You need the frequency of buses and trains far better
You need to make more trams
Some of this is now in Andy Burnham's power - when he was elected Manchester got the power to take control of the buses (currently it's a free for all.). But fixing this mess is going to take years. And cash. The benefits are plain to see, but will people stump up the money?
When I bought my first car when I left London, I was keen that I didn't just use it because I had it when I could take public transport. But boy, it's not made easy.