This is the order I'd go with:

Carpinter - Byregot's - Use Careful Synthisis 1 for the time being.
Weaver - CS2 - Most used progress skill.
Blacksmith - Ing 2 - Helps a lot of 1/2 star crafts.
Alchimist - CZ - Extra CP
Goldsmith - Inno - Improves quality
Armoursmith - PbP - Only really used for 3 star crafts or 2 star+ furniature, it is a very good skill, but needed only once your on higher tier stuff.
Cullinary - Reclaim - Can be handy, but not really recomended.
Leatherworker - WN2 - Skill with very few uses, you'd really never miss it. ***this would be in 3rd/4th place if you need it for crating your own AF sets! a lot of useful recipes***