You can donate anything you want, and I believe it increases your favor with the Giant, which increases your exp and will (eventually) allow you to skill faster. ^^
You can donate for favor...each shrine gives back a little differently...you can also choose supplication which gets you a ticket...So far my tixs have offered teleportation to other streets But I am new at this too...so a more skilled player will probably insert facts for us...
Each is devoted to a different god/giant – you can see which one to by clicking on 'get info' link when hovering over the shrine (to get an idea of what favor will be given to you)
And I guess they can also give you a music box.
Apparently the effect depends on the value of donation and there are even special powders that serve as a high-value gift. alpha.glitch.com/items/464/
Using a hallowed powder can greatly increase the benefit of a donation. Large xp gains can be made if the right item is donated after use of such a powder. :)