I think the word you are looking for is
Come together singing let it be,
Referencing songs from The Beatles - a musical band. 'Come together' also puts us in mind of 'band together'.
And gather that which is free,
A band is a thin strip of material which can be used to tie things together, e.g, a hairband.
As a child a simple weapon,
I think this refers to an elastic band which can be used in children's weapons like a slingshot or toy gun.
To best not be in possession,
This could be either banned (a homophone) or contraband.
Follow so you can ride with me,
Let us play and move exactly,
A no-good-doer finishes it,
Putting 'it' on he end of band creates bandit - a no-good-doer
Forging not ahead makes your commit,
I think this is a play on words and refers to a wedding band (wedding ring). 'Forging not' sounds like 'tying the knot' as in 'wedding' and putting it ahead of band gives the result, a sign of commitment.
A mystery is all I can see,
Bandanna (as suggested by OP)
As you dress me up so nicely.
Bandages (as suggested by OP)