I am going to be the ghetto crafter on this thread...excuse me while I dab!
I leveled up all things from 50-60 w/in 2 weeks of HW launch...didn't have Mog Quests, didn't keep my gear updated, only used what I was awarded through class quests until I could make the HQ chimera stuff, only used Bouli HQ through the process. Now I did start with a full artisan set and Lucis tools but those were considered obsolete by 52.
Now I did the GC turnins, in my opinion these are a MUST, in some instances you can get over 2 mil xp for an item.
Level everything together because you will get skills in one class while you try to HQ and try to make the other stuff HQ in the process for leves.
You literally have a stocked MB now which I didn't have when I lvled, that is why i lvled gathering with my HW crafting b/c I was not waiting on anyone. So this is an advantage for you.
I will echo that I didn't use cross class skills until I got to 60 so I wouldn't worry about them at all.