I spent about three years contracting, and the way I started was to ask the company I was a permanent employee at (who routinely hired contractors) if they would like to hire me as a contractor instead of as a perm. They said "OK" and so I quit as a perm and restarted as a contractor (I actually also managed to negotiate a four day week and one of the four days at home, back when remote working was much less of a thing, for effectively about twice the money - man, I really knocked that negotiation out of the park, in retrospect); I spent about three months there before moving onto another contracting gig, which I think I got through a recruiter. I know a few other people who have done exactly the same thing as their initial contract.

Derek has mentioned basically all of the positives of contracting - the downsides are that shorter contracts (three months, or anything less) tend to be either quite dull (very routine work that nobody else has the time for) or unpleasantly intensive (desperately trying to ship a disaster); in some organisations you will be treated as "just another bloody contractor" by the perms, who will know you make more money than they do and hate you for it; you may get less responsibility than you would have as a perm, or generally "less say"; and if you end up contracting for the sort of company that you'd hate working for as a perm, it will be just as bad as a contractor, except you'll probably care even less for them because you will know that you can quit very easily, which is, itself, a bit demotivating.

I transitioned from contracting to running my own consultancy, which is much harder work, much more stressful, and with a lot more risk, but quite a bit more rewarding (for me personally - definitely not something I'd recommend to everyone).