There are many different ways to experience Elysium depending on your style of play.

Explore
First thing's first, get situated, learn where you are, then explore around you. Don't forget a light source for wandering in the dark, and watch out for rivers! Check out what's around you on the Explore page
Serve a city
Run a shop or maintain some industries. Prove yourself and maybe become a minister of your city, perhaps even an emperor someday!
Join a guild
Guilds offer skills that are similar to each other with abilities to be taught with fewer lessons than if you were to learn a skill out of guild.
Questing
Quests are given either from your city's court advisor, or different NPCs throughout Elysium, each with their own rewards. Do be prepared to sail for a number of these quests, don't be afraid to ask for help!
Crafting
Crafting goes beyond weapons and armor. Furniture can be created, jewelry, clothing, more complex foods. Not all items are created through a crafting skill, some are created by industries, and other means.
Sail the world!
There are more islands than just the three major continents, don't forget to pack for the trip though. If you need a ship, give a shout and try to buy one from a city. Hint: ask Darkfly, he's a pretty cool guy and can sell ships. Watch out for sea monsters!
Search, treasure dig, treasure dive, and help out distressed citizens
There are a number of ways to simply find items. Grab a spade and dig on the roads to find stolen items or search cities to find anything from an apple core to valuable silk items, and more. You can also help distressed citizens in cities all over for random gifts, just listen to the citizens speak to find out who needs help next
Common skills
Pick up skills that don't count toward your core skills. Learn more advanced cooking with Cookery, learn to sew worn out clothing, build up mana reserves, and other skills.