1. Be amazing at your instrument, and the more instruments the better. You need creativity as well as technical prowess. Actually the most important thing for session work is probably timing - practise to a metronome. Also you definitely need pro quality top-of-the-range instruments/gear to get taken seriously by the good studios.
2. Get some sort of showreel/portfolio and/or website/youtube/whatever showing off your skills. Include as many genres/styles as you can, and also experience and skills (eg can you sight-read music? etc)
3. Call all the nearby recording studios and ask if they can add you to their list of session musicians (probably they'll tell you to email a portfolio, contact details and rates etc)
4. Wait for the call... whenever a project comes into the studios you've called that requires a session player of your instrument, they will call the best/cheapest/closest depending on what the client wants. If that's you, then you'll get the work.
A tip would be to charge as little as possible until you have done a few sessions, then slowly raise prices. Have an hourly rate and a cheaper day rate eg. £10 per hour, £70 per day.
That's how I'd suggest getting into studio session work. Many people would say you'd need to move to London, but there's quite a few studios around Yorkshire (where I'm from), and everywhere else (wherever you're from), and probably less competition than there would be in London. You might get 1 or 2 bookings a month at first (assuming you call as many studios as possible), hopefully more once the studios know you and your reputation.
Regarding live session work, I'd probably do the same but call management agencies instead of recording studios. I assume its management who sorts that stuff out for solo artists...