Problem while shifting gear from first to second

[Fazerider]

Clutch adjustment may not be the cause of this problem: if there's too much free play (so the plates don't completely disengage) then difficulty changing from neutral to first is usually more noticeable.
Second gear is a known problem area on the FZS600, but if yours has only just started suffering, you may be able to improve it by adjusting the position of the gearlever slightly... lowering it a few millimetres may help you make the shift more positive.
Beyond that though, some have had success with new return and detent springs on the gearlever mechanism (the bit under the clutch cover), otherwise it's a change of selector forks, perhaps the selector drum too ... and possibly even some gears themselves if the dogs have become sufficiently damaged.