Leveling Strategies And Options 

Islands
Although the lowest experience gain for the energy, weak pets can defeat islands easily. Monsters give exp on the islands, including guardians who give double the usual. On ten energy a pet can get a half to a couple levels depending on how high level they are on entry. Reference the island guide overview to see which islands are best for leveling.  

Crafting
Crafting recipies, golems, buildings, ect, give experience. While a nice amount, it is not a reliable way to level a pet and takes resources and time. 

Quests
Sending a pet on a quest gives it experience upon return depending on the difficuly of the quest (copper, silver, gold). Although more reliable than crafting, it is still a slow and infrequent way to level a pet who may be missing requirments.  

Battlefield
The battlefield is by far the fastest and most energy effiecent way to level a pet as long as it is capable of battle. The opponet's genetic power and genetics do not give more or less experience. A pet can battle opponets that are within five levels lower and higher. More exp is gained for defeating a higher level opponet, but it is also riskier and easier to lose.

Riding Hall
The riding hall is most useful at uppers levels (40 - 50) when gaining exp on islands becomes less worth it and arena opponets become tougher. It costs silver to train your pet, the further along they are to the next level the less the cost will be. As in, a pet at 0% to level 50 will cost more than one at 25% to 50.
To have access to a riding hall you must first build one and wait for its completion. Young, busy, and level 50 pets can not be trained. If your pet is available for training you will see a green button with the word "train" on it, next to their level and under their name. The cost will be alongside it. Training takes four hours for each session, and each session takes the pet one level higher.

Base Riding Hall Silver Cost For All Levels

Level 1: 80
Level 2: 160
Level 3: 240
Level 4: 320
Level 5: 400
Level 6: 480
Level 7: 560
Level 8: 640
Level 9: 720
Level 10: 800
Level 11: 880
Level 12: 960
Level 13: 1040
Level 14: 1120
Level 15: 1200
Level 16: 1280
Level 17: 1360
Level 18: 1440
Level 19: 1520
Level 20: 1600
Level 21: 1680
Level 22: 1760
Level 23: 1840
Level 24: 1920
Level 25: 2000
Level 26: 2080
Level 27: 2160
Level 28: 2240
Level 29: 2320
Level 30: 2400
Level 31: 2480
Level 32: 2560
Level 33: 2640
Level 34: 2720
Level 35: 2800
Level 36: 2880
Level 37: 2960
Level 38: 3040
Level 39: 3120
Level 40: 3200
Level 41: 3280
Level 42: 3360
Level 43: 3440
Level 44: 3520
Level 45: 3600
Level 46: 3680
Level 47: 3760
Level 48: 3840
Level 49: 3920
Level 50: NA

Total: 98,000

With Trainer Invite: 49,000

Hope you don't mind me referencing this thread in, and adding a bit of the information from here, to a google sheets:

https://docs.google.com/spreadsheets/d/1zitdrEvtuVTHkfxFM0IZXZ3phUL1P8AY1BmAtqKw2q8/edit#gid=0

It has the Experience gained for winning, only Arena though (I don't explore with anything besides level 50 now, and didn't track the Exp.G there), and total material required for all current buildings -- though I added the cost for making items, such as carpets and banners to it too.