Want to earn quick experience points for your monsters without having to power them up and sacrifice other monsters? Build the Tranquil Forest, the Crystal Lake and the Gusty Cliffs when you unlock them. The Tranquil Forest is, unfortunately, the only one which can be purchases with Mana, so you’ll have to wait until you rank real high for awhile in the Arena to afford the other two. You can place monsters inside them and they will earn experience points every hour.
Want to increase your monster’s skills? Go to the power up area and boost your monster with an identical monster. Identical monsters of different elements are okay. This will boost your monster’s skills even if your monster’s level is already maxed out.
Devilmon, which can be found in the Cairos Dungeon (again), can also boost skills when they are used as materials for boosting a card. They can work with any card. Angelmon will give loads of experience to a card of the same element as they are. Rainbow Angelmon will give loads of experience to any card that they are used on.
The process for evolving a monster is first, maximize the experience level of one of your monsters. Then, get monsters of equal rarity in stars – the number of monsters you need is equivalent to the star rating. Then, use those monsters (for example, four level-4 monsters to evolve a level 4 monster) as material, and your monster will go back down to level 1, gain one star in rarity, and have a new form.
Summoning pieces can also only be found in Cairos Dungeon (starting to see a pattern here?). Summoning pieces are specific to one monster, and once you get enough summoning pieces of the specific monster, you will be able to summon it in the summonhenge. Different summoning pieces have different rarities.
Check the item drop info lists for every dungeon to see what you have a chance of finding. All of them, though, except for Cairos Dungeon, will only show what monsters you can find, and you have a better shot at finding the 2- and 3-star monsters when you play them on Hard and Hell modes. You also have a better shot at finding runes, mystical scrolls, multiple unknown scrolls at a time, and other goodies when you play on harder modes.