You have to complete the Main Storyline. So if you are really determined to get to Dragonspyre you need to stick to the main quest and not do side quests. Here is a link to the Quest Tree for Krokotopia, once you beat Krokopatra you can go on to Marleybone and from there Mooshu, Dragonspyre etc. There are also some side worlds such as Wysteria, Aquila and Grizzleheim. You usually get them as a quest from Ambrose or someone their main requirement is you must be a certain level. I think 20 for Wysteria and Grizzleheim and 30 for Aquila.
To get to other worlds, you have to complete the world you're already in. You need to finish the main quest in Krokotopia to move on to the next world. At level 20 and 25 there are side worlds that you can go to, but after you finish them you can't go anywhere else. Those are the only ways to go to other places
In order to access other worlds in the Spiral, you must first complete all the main quests in the world you're currently in. For example, you have already completed all the main quests in Wizard City, which granted you access to Krokotopia (The world you are currently in). You do not have to do the side quests in order to progress to the next world, but it would give you a big head start in the next world if you were to do so.
If you are having trouble distinguishing the main quests from the side quests, then I'd recommend going to the Wizard101 Wiki. Not only do they have information on all of the quests in Wizard101, but they also have information about equipment, pets, mounts, worlds, monsters, NPCs... Nearly everything in the game! It's a very useful tool to have!
Continue what you're doing. Wizard101 Worlds are level-based. Once you finish Krokotopia, you will be able to move onto Marleybone, and once you finish Marleybone, you will be able to move onto Mooshu, etc.
The Storyline goes: Wizard City > Krokotopia > Marleybone > Mooshu > Dragonspyre > Celestia > Zafaria > Avalon > Azteca > Khrysalis > Polaris. Grizzlehiem and Wysteria are optional side worlds, but I reccomend you do them.
After Krokotopia, Marleybone After Marleybone, Mooshu After Mooshu, Dragonspyre After Dragonspyre, Celestia After Celestia, Zafaria After Zafaria, Avalon After Avalon, Azteca After Azteca, Khrysalis After Khrysalis, Polaris And hopefully there are many more to come 110 110 110 52 34 31
You need to continue with the storyline. You can't go to Dragonspyre yet because you don't have the spiral key, and the world of Mooshu has the key. You don't have the key to Mooshu, so you need to go to Marlybone to get it. You don't have the spiral key to Marlybone, so you have to go to Krokotopia for it. Also the Order of the Fang and the Krokonomican are main things in Krokotopia.