How Does Trading Work/What Items Are Worth Saving?

What items are worth saving is a very difficult question in PoE and depends on the league, how long that league ran and how far ahead you are in comparison to the majority of the player base.

As an example: In Standard you are not likely to sell any yellow items until they are really, really good. That league has accumulated items from many thousand players over several years and dying doesn't matter until you are a very high level. Trade currency or do the leg work for rich players.

In a new league (3-4 days old) you can often sell semi-decent gear. Got some boots with speed and resists that you don't need? Put them on the market. You can even sell leveling uniques that are nearly worthless in Standard. After 2 or 3 weeks the market starts to saturate and you can only sell good gear as most players have some backup items of their own to level new characters. It gets gradually worse the longer the league has been running.

In general what are you looking for:
a good life roll, 2/3 high resist rolls and preferably high ar/ev
energy shield items can get away without life roll if they have really high ES values
weapons are more diverse, depending on your build, hard to give advice

Personally I would recommend to start playing without any thought to trade unless you need to buy a specific item for your character. Once you have played a couple of different characters you will develop a decent idea of what good item are and what's vendor trash.

Good luck!

Also: Don't play hardcore as a new player. There are too many nasty surprises waiting for you that you don't expect. I'm looking at you, Dominus!