Experience is what will increase your level to unlock new items in the shop. 
Plowing, whether done with the basic plow tool, the tiki plow, or the mega plow gives 1 xp for each plot plowed. Seeding, regardless of which seed or which tool is used to seed it, gives 2 xp. The thing that makes each crop more or less worth it is the harvest experience. The shorter the time period for ripening, the better the crop. Taro Root is the best crop from the shop that is free, followed by Raspberries. The Lilikoi is an even better crop than either of those, and Dragonfruit, a 2 hour seasonal crop was best of all.
An example of fertilizer use to boost experience is to seed a field of 1 day crops, apply the 1 day fertilizer, and immediately have your crops ready for harvest. Keep in mind that there is a daily xp cap.
Animals that have shorter harvest periods will give more experience over the long run with all factors considered. A Turtle may sound attractive, but if something happens during the five days while you wait for it to be harvestable and it ends up in storage, you will lose those days and have to start over. On the other hand, animals such as the Brown Cow, English Game Hen, Kiwi and Cassowary are much lower risk. Even if they are displaced into storage and their harvest time reset, they will take less time to become harvestable again.
The most experience efficient animal is L'il Lagoon Lucy (11xp a day), followed by the Chipmunk (9.6xp a day), Koala Bear (9xp a day), White Wolf and Beaver (6xp a day) and Monitor Lizard (5xp a day). The Brown Spotshire Pig, Armadillo, and Black Cat all give 4 xp a day.
Brown Cows, Seagulls and English Game Hens give you a total of 3xp a day if you manage to get all 3 harvests in a day. Not to mention the fact that the harvests from Brown Cows and English Game Hens can be used in Cooking for more experience! Squirrels also give 3xp a day.
Brushing your animals is also a great idea, since each time you brush your animal, you gain 1xp and the animal gains 1xp.
The Breadfruit Tree, Apricot Tree, Pine Tree, Ceiba Tree, Rambutan Tree and Mangosteen Tree are currently the best for gathering quick experience (4 XP/day), followed by the Rubber Tree (3.75 XP/day). The Chinese Juniper Tree (available through gifting), Cherry Blossom (available with MC) and the Peach Tree all give 2 XP/day. See the Trees page for a complete listing of trees and their associated XPPD (XP Per Day).
Of course, basic farming only takes up a tiny portion of what makes players level up quickly.
Fishing is not a great way to make coins, but the xp given from it is useful if players do not have the time to go stealing or salvaging from neighbors' islands. All fish nets that are retrieved will give 1 xp. Fish that escape will give no xp and result in a loss of coins. If you are going to leave your computer for less than 10 minutes, it is recommended to cast your nets and when you come back they'll be ready for retrieval. Upwards of 10 minutes it is not recommended to cast your nets, although putting a recipe on to cook (see below) is still recommended.
Salvaging items from your own island or your neighbors' islands is a way to make some experience. Any items salvaged will give 1 xp, although the number of coins will vary depending on the item salvaged.
Stealing crops, fruit, or products of neighbors' crops, trees and animals will give 1 xp regardless of the value of items stolen. With the addition of cooking, it is recommended to steal what ingredients you are missing to cook what you need.
Whenever you set an item to cook, you will gain 1 xp. Cooking increases the value of your crops, which is useful as you can then use the coins to generate more xp. Cooking short recipes is the best way to generate experience. Synchronizing it with your Fishing holes will make the task of cooking even easier.
What many players forget is, whatever you set to cook will not burn or rot. You could leave it 10 minutes past the time it is finished or 10 days (not a particularly wise idea, but it still won't burn).
If you know you are going to be away from your computer for about 3-5 minutes, toss a 3-5 minute recipe onto a stove and cast your fishnets. When you come back, they'll be waiting and you can collect them for coins and experience.
If you are leaving for more than 5 minutes, put a longer recipe on to cook - but don't cast your fishnets or the fish will escape by the time you get back.
With the addition of "excellent" recipes, the experience gained from cooking increases. Each time you gain a star, you gain experience bonuses and a higher chance of making a recipe that gives an additional experience point.
All items that are purchased from the shop with the exception of crops, Island Upgrades, clothes and Boosts give 1 xp for every 100 coins spent, rounded down. So for instance, if you were to purchase Red Roses, you would get 12 xp although you spent 1250 c. Experience gained from purchasing items is limited to 1000 xp per item. For example, when you purchase a Dinosaur Exhibit for 160000 coins, you receive only 1000 xp, not 1600.
If you plan to gain experience by buying and immediately reselling an item, this results in a loss of 95% of the coins spent; it is still a viable strategy regardless. The best items to purchase for this strategy are the Crate and Stick Post. Both cost 50c each, but give 1exp instead of the expected 0 since all items must give at least 1. Buying these items enables a player to earn double experience per coin spent than other items give, meaning every 100 coins spent will give 2 xp.
Crops will give 2 xp when seeded regardless of value. Island Upgrades, clothes and Boosts give no experience.
Any and all items bought with Meteor Credits will give 1 xp upon purchase, regardless of the price unless it falls under the categories that don't give experience.
Finding your neighbors' Lost Cats and Lost Dogs or researching their Recipes will give you 20 xp. After the animals found, make sure to remember to go pet them for another 20 xp. When you find you own dog and cat, make sure to post the petting link so neighbors can pet your dog and cat for 20 xp, while giving you 20 xp at the same time, making a win-win situation.
After posting your own dog or cat petting link, it is recommended to go to that post, right click over the "Pet the dog/cat", and select "Copy link address" (exact wording will vary depending upon your browser). Open a Notepad or Word Document, and paste it there (Ctrl+v). Save it and occasionally repost it for some additional experience as neighbors click it.
Collecting the bonuses from friends who are gaining stars for their recipes is also an excellent way of quickly gaining experience. You can collect 10xp, 25xp, or 50xp, depending if it is the first, second or third star they've gained.
It might not seem like it, but even if you get gifts you don't really want from your neighbors, you can sell them and use the coins to purchase items for even more experience. The most profitable gift to sell is a Hammock because it sells for 500 coins.
Make sure to return as many of your neighbors gifts as possible however, or some will feel offended and refuse to continue gifting you!
Other Important Things to Keep in Mind
The number of neighbors you have will significantly impact your ability to level up without the use of boosts. More neighbors means more islands to steal and salvage from. More neighbors means more dogs and cats to be found and petted. More neighbors means more crops to revive. More neighbors will impact how many gifts you can get in a day.
While purchasing with real life money is the easiest way to get your Meteor Credits, it is possible to complete offers and get your Meteor Credits that way.
The amount of coins you have can also impact your ability to level up quickly. Buying good investments such as a Tree over a decoration or flower that produces nothing is key to leveling up quickly.
There exists a daily xp cap, as well as an overall XP Cap, which is currently at level 95.
The size of your island will impact the number of crops and trees you can plant. Buying a larger island the moment it is available may seem attractive, but if you can't afford to plow and seed your new plots, then waiting until you can is a good idea.
While it is possible to cheat your way up levels, such behavior will result in a ban. So unless you don't plan to play Island Paradise ever again, don't cheat.