I started by tagging along with a friend, who recommended me for a project he was already working for. It was easier to find the courage that way.
The next time somebody hired my, I simply said I'd prefer to work on a freelance basis. They were OK with it, however, because of my lack of experience with freelancing, I accepted a rate that was much too low. So that may be an easy way into becoming a consultant: just say you want to be a consultant...
I figured that out when I became I'll - nothing serious, but I realized I need to ask for more money to compensate for downtimes and risks that employees don't have (also, saving for my own retirement).
After that, I also registered with some freelance sites, like Jobserve and Gulp (latter might be a German thing, but they offered nice statistics for average salaries).
Because I struggle with occasional burnout or simply hating my job, I din't even do much of the network building or friend recommendations. But recruiters keep calling, just as long your CV has the right keywords. Even long gaps of "unemployment" don't seem to matter that much.
However, for the same reason, these days I would almost recommend against sending out your CV or registering with recruiting agencies. Now my profile is out there, and I can not retract it very easily. So I keep getting those calls, even though I don't really want them anymore. I'd say if you can get by without sending out your CV, it would be better.
I haven't really cut ties completely - maybe if I told those recruiters that I definitely don't want to be called ever again, they would comply. But I wouldn't count on it - in general, they just ignore "wishes" in the CV, like preferences for locations to work in.
I was also that way probably not necessarily getting the best jobs (in terms of interestingness, the pay was always OK). It was more "crap, my bank account is running low, I better accept this contract", rather than strategically working towards work I would like. A bit of a "golden handcuffs" problem, I guess - you have to say no to good offers to be available for the ones you actually want.
This may have been more serious for me than it sounds - by being stuck in dreadful Java Enterprise projects, job frustration would mount again, causing me to quit for months, causing me to eventually accept the next best offer, for the cycle to repeat. Maybe with a more proactive way of acquiring contracts, the frustration cycle could have been avoided altogether. (Not sure, though).
Nevertheless, overall I think consulting is great in principle. You earn more money, see more different companies and projects, and the hiring process is not that fucked up. Usually they give you an initial contract for a month, to see if you work out. So they don't have to evaluate you for weeks and weeks to see if you are a good fit. If you are not, they simply don't extend your contract.