go to https://dev-na.ffxiv.com/
Once you are logged in on the left hand side read the beta manual. It will tell you everything you need to know.
However in order to get more quests you must go to the corresponding cities guildleve counter. I haven't been to Gridania yet myself, but it should be near the center of the city with green/blue glowing stained glass above the counter. From there you are able to get more leves.
In addition you must remember this is open beta, and many of the NPC style quests are not active yet. Play around with some of the other mechanics if you are unable to actually progress your character. You will not be able to keep this character anyway.
Hope that helps.