The first place I'd visit would be Galadriel's house. Giggity.

After finishing to your introduction to Middle Earth in Lord of the Rings Online, the first place you visit as an Elf is Falathlorn. Learn about the quests and ruins of this sylvan land with Darkgolem's newest guide!

There are also the fields of Neth Hilith (located east of
Celondim), Tham Gelair (a ruin, infested with wolves and oversized
dragonflies called "Pests", Limael's Vineyard (overrun with goblins),
The Eastway (just the entrance to the Shire, found east of Duillond,
across the river Lune), and Dol Ringwest (another ruin, infested with
goblins).

Check the guide out at Ten Ton Hammer's LoTRO site.