First and foremost, I hope you've been getting the Dragon Trade Pouches from Master Hong whenever they pop up to increase inventory space as that'll be highly useful.
As for utilizing/storing items...
Basic Storage
This storage can hold pretty much anything and is the most versatile of your storage tabs. However, it's also the hardest to upgrade and therefore the smallest. Equipable items (weapons, accessories, gems) you're not using are best placed in here. Pretty much, anything you can't put into your...
Secondary Storage
This is the easiest one to upgrade and, though it does limit what kind of items can be put into it, the Secondary Storage will hold the bulk of your items. Primarily, this is where materials and consumables you're not using ought to be stored. Anything you don't need to have on you all the time should go in here as much as possible.
Premium Storage
If you're a premium user (or luck out and get a 7 day premium from an event or giveaway) this storage is perfect for items you're saving for a rainy day. While you have premium, you can access it like any other storage, but if your premium runs out, you won't be able to put items into it. You CAN, however, take things out or leave them in there. That's why this is great for things you want to hang onto, but don't plan on pulling out until they're being taken out for good. For example, if you want to fuse some badges together and know it'll be a while until you get the materials, you can stuff the badges into the premium storage until you're ready to fuse and equip them.
There are other handy ways to store some items...
Showroom
If you have outfits or weapons you want to store, most of the time, they can be put into the Storeroom. This should especially be the case with outfits as there's a LOT of them.
Received Items
Notice the little box symbol at the bottom of the screen? Notice how whenever you do a Daily Dash spin, the item you got from the spin is shown near that box in a little message? This is the Received Items tab in your mail. Pretty much every time you get something from the game that is going to your account and not directly to a character (IE: when you buy something in the Hongmoon Store), it will arrive in your Received Items. Now, this isn't a storage in the same way as the other above items (as in you can't put things into it), but you can leave any items you don't intend on using in there. The added benefit of this is that Received Items is across your entire account, so you can pull items out of it when you need them on any character you need them on. There is no real limit to how many things you can leave in there as well (I'm up to 14 tabs myself lol), so it's a great way to hold onto items until you actually require them without taking up inventory space.
So, with all that in mind, I hope you have a general idea on what to do with your inventories.