Can someone help me with strategies for Sky Ruins, Dragon Gate and Apex Mountain? I'm not sure how to complete these efficiently and any advice will be appreciated.
Of course what worked for me may not work for you, but here's what I did:
For Sky Ruins I used a team of Mewtwo, Charizard, and Palkia. Use team attacks against one of the dragons and have Mewtwo use Psychic, Charizard use Heatwave, and Palkia use Spatial Rend. Mewtwo's Psychic will lower the target's Special Defense over time, causing it to take damage. Once you beat the first, the second will have already been weakened by Charizard and Palkia and should go down somewhat fast. If you don't have Charizard, any powerful move that hits the entire room and inflicts special damage is good as well.
For Dragon Gate I actually beat it using a team of my most powerful Dragon-Types, which was Haxorus, Hydreigon, and Giratina. Haxorus is especially good if you abuse the combination of Swords Dance and Dual Chop. Use two Swords Dance and Dual Chop will easily be inflicting 100+ damage per hit, and that's multiplied by two because it's two hits. Things get even better if you throw Emeras into the mix, especially Barrage, allowing for up to four hits, and Power Boost X/Y for even more damage.
I don't really have any suggestions for Apex Mountain, as I beat it with relative ease and no strategy. I just used my two starters (Charizard and Greninja) as well as a Palkia. My two starters had been stat boosted simply using the vitamins I've found throughout the game, not even buying any.
Of course, a generous quantity of (Tiny) Reviver Seeds and Oran Berries is always nice as well. Also very helpful is Truant Wands. Bring two for the Sky Ruins and three for the Dragon's Gate. The opponent not being able to attack every other turn makes a big difference.