Yeah, there are a lot of quests, usually:
- At least three big quest lines in each zone that will give you a skill point
- Every 5th level there will be a quest in the main quest line (starting at the anchorage) that will give you a skill point
- A lot of side quests
If you do all the quests in a zone you will out-level the zone (and this will make the next zone less interesting). The point of all the quests is to have more choices, I guess. You should probably try to do all the skillpoint quests and the main quest as you will be needing many skill points early on. Here is a list of the main quests for Aldmeri, for example, you can the same overview of the other factions as well: http://www.uesp.net/wiki/Online:Aldmeri_Dominion#Quests
This is a good add-on for finding sky shards: http://www.esoui.com/downloads/info128-SkyShards.html
You can expand your inventory and bank space 10 slots at a time with ingame gold in increasing amounts. The crown store offers some stuff to speed up some things (like increasing riding skill and bank/inventory space upgrades) but ultimately you can get it all ingame without spending any real money (just takes more time). I would only consider buying it fromt he crown store if you really wanted to jump start a character and there is a very good crown sale on.
So where do i go to increase inventory space?