I did consultancy for 5 years, decided I liked a fixed position better after that.
Seeing lots of projects can be fun, but you're never really at home anywhere. You rarely have the satisfaction of seeing a project through to completion, you're almost always called in as the hitman to fix a problem or temporarilly replace someone who suddenly became unavailable (death, longterm illness, not the nicest reasons to enter a team).
I had a permanent position with a consultancy firm, who paid me even when they had no work for me. Best of both worlds really, and not something that's done a lot these days.
If you can get that, and you have a good employer,
you should get ample training because it increases your value to your employer (and thus the price they can charge their customers for your services).
But many contracting firms will ignore employee training, instead milking you for as many hours on-site with customers as possible until you're useless because you lack training in the technologies that are in demand.
At that time you'll be dumped, either just discarded like an old pair of shoes (fired...) or find yourself pushed into deadend jobs with no job satisfaction and given trouble everywhere in an attempt to get you to quit.