I am new and trying to understand.

Should I quest.

I finished the season so now I am lvl61.

I have only finised a few quests.

Should I quest now? What say you?

First time through you should absolutely be doing all the main quests, and a few quest lines that used to be part of main as well (Valencia  2, Atoraxxion, others?). Not only for the lore and rewards (many of those aren't really relevant anymore, but some are), but for the contribution  points and the fact that those quests open up access to other things (more quests leading to more rewards, some game features).  Also take a look at the "suggested" quests in the quest menu ("o" key). There are various family once-only quest lines that open up game features and explore game-lore more. Meanwhile, you'll be getting more "knowledge" (the game thing - "h" menu) that translates to "energy" and also more access.

Sure, it's a time investment. I recommend at least one full run-through without r-spamming / skipping quest test / audio / cut-scenes. 

What the above poster said about certain quests opening up access to things is accurate. The Magnus quest is a must if you haven't done that already, really no debate on that one . LOML quests help with the Black Shrine Boss stuff.  The Rebellious Spirit Crystal is a very good item to have as well that you can craft or purchase. If you purchase it I think it's around just under 2 billion silver. In order to craft it you need the materials from the three main quest lines, I think they are Mediah, Valencia and Kamasylvia, each quest gives you one of three required materials.  Just a few things that quests can help you with. 

I am enjoying the quests. I will try to do all the quests. I may have thrown away some quest rewards.

I was full and had to empty some boxes. Thank you for the pointers.

Where to put stuff... we all become in-game hoarders :) There's probably guides around, but here's a few ideas:

Early on I found it useful to put what I could in the "warehouse" (aka "CM", or central market).  Not to actually sell anything, but because of its much greater slot limit. Not everything can be put there, but it can help. CM weight capacity can be increased some with additional "family fame", but you start out with quite a bit already.

Once you have the Magnus quests completed you'll have easier access to storage in other regions from any storage keeper NPC.  That will help with more items. You can expand storage a bit by investing contribution points in some houses in towns that have storage.

For more character inventory space (separately for each character), the main quests give some, and there are some "suggested" quests that give some more.

Some people use wagons and maybe boats to store a few extra things.  However, the item find feature (ctrl-f) doesn't see those.

And there is also specialized "storage" for things like artifacts and lightstones.

You'll find that some items only stack in the CM, such as crystals, seeds, and accessories (and some other things).

And to work with storage and warehouse from any location, not just storage and market NPCs, you can get "maids". Those are sometimes given away during events, or can be purchased from the pearl shop. 

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Oh, additional "questing" tip: check out the "progression pass" - an up-arrow icon at the top, to the left of the mini-map. And of course, there's the season pass for a season character, you probably already completed that.