Inventory management is truly an issue in this game, but it's not an impossible task. I mean, here's an hour long video on it.

First off, a lot of it is literally there to push you into buying bank expansions and storage expanders in the gem store. Creating an issue and selling the remedy is just part of the game's monetization scheme.

For a budget version, dumping stuff onto spare characters is not a bad idea either.

Now that's been said, let's go through the basic inventory management loop:

1. Unalive enemies and inherit their possessions
2. Open loot bags (Pro tip: "o o" works great as an inventory filter for these)
3. Identify unidentified gear
4. Salvage all (Basic for blue/green, Master/Mystic for yellow/orange)
5. Deposit materials

But that won't work unless you got the space, so let's trash some stuff.

1. Anything that says "Trophy" is an event or heart item and can be destroyed.
2. Crafting materials that don't fit into your material storage might just fit into someone else's, so you can sell them on the trading post. For best results, sell the full stacks from your storage so you can deposit again.
3. Sell things like sigils, runes and crests. Preferably on the TP, but vendor if the price is terrible.
4. You're not reading that book you got from am achievement ever again, trash it (or swipe your credit card, because there's a lot of these)
5. For anything else, check the wiki. You can easily do that from the game by typing "/wiki ", then Shift-click on the item and press enter.