Is it good to do management consulting being a part of any consulting firm or become an independent consultant right away?
Being a part of a Big4Company. It grants projects instead of chasing customers/prospects.
Skill is mind search conclusion and knowledge is work experience.
I think both working have their own positives and negatives. But starting one's consulting career with even a smaller firm makes a big difference in one's consulting mindset and process learning. Working for a couple of years in a firm, building contacts & experience and then taking up individual assignments will be more helpful in a consulting career.
You can check guide about this topic in Cleveroad blog.