You're in luck! The moogle quests they just added take the place in HS for the old Ehcatls(mostly) in that they are DoH related and give you xp for doing them. Other than that, leves and log like normal, as well as the daily hand ins at your Grand Company place. DoH/DoL also have an entire collectible system for turning in items for "scrips" where you use an ability that attempts to gather/create an item that can be turned in for XP and at 55+ blue scrips, which are used to purchase related gear from vendors in Idyllshire. Without going into a ton of detail here, an easy place to check out guides (not really shilling, but if someone has already done all the legwork, why type it all out~)
Crafting Archives FFXIV Guildand then pick the class from the tabs at the top for a quick guide on the best leves/collectible as well as some generic gear advice and location information.
As for order to make them easier? The only cross-class ability that you will need/use beyond the ones you already have for the most part is Makers Mark (GSM 54) and perhaps Muscle Memory (CUL 54) but really? The built in abilities you learn in each class that can't be transferred to others are plenty for leveling and you can basically do them in any order aside from the cross class stuff you already have access to. Careful Synth II/Comfort Zone/Byregot's Blessing/Tricks of the Trade/Ingen I or II/Steady Hand II and Flawless Synth(for use with Maker's Mark, if you go that route) are really the most useful cross class abilities for leveling. It really is just business as normal, except now you have the wonder that is Precise Touch for all DoH at 52.
DoL is the same, and given the inflation of the servers in general will make leveling much less expensive and painful for your DoH.
As you do the HS quests, you'll get these 300k bonus xp for 20m items. Until you get to 55/56 you generally won't be able to really use all of that xp up in the time limit, so it is better to hold onto them for when you are mass producing leve items or what have you after you are to that range.
Edit - I will add, however, that if you are trying to supply yourself with all the mats from gathering classes, you are almost 100% going to hit cap with your gatherers before your crafters. HS mat collection is substantially more volume intensive than ARR, especially for any of the metal items. I would generally recommend that you save any leves for the DoH unless you are about to cap out or something.