You can go into battle and fight manually, but there are two ways that you can speed up battles for rapid experience gain. One is to go to a level that you have beaten previously and sweep it. You can sweep it as long as you sweep tickets that you can use, and you'll beat it instantly. Another is to hit the "auto" button in a level. The game will speed up and your character will fight automatically.
Characters and Slots
You can make up to two characters at a time, with two new slots being available at a price of 100 Zen apiece. You will only have to go through the tutorial once, with your first character, though. After that, try the other characters and then decide whose attacks you like the best. Even though the game says some are intermediate, beginner, advanced, etc., it all depends on one's personal preference.
Extra skill points
Every time that you gain a level, you'll earn skill points in addition to the stat boosts. Go to the skill screen to unlock new skills and to upgrade your existing skills, both active skills and passive skills, as soon as possible in order to get an edge over other players and over the enemies that you have to kill. Passive stats are especially important as they will increase your base stats no matter what you have equipped.
Sell Stuff and Chat
Talk to everyone that you can inside of the town. You can't talk to most people as they're usually other players, but you can talk to anyone who has a conversation bubble over their head when you get close to them. Simply tap on them to talk to them. Some will have tips for you, and some will want to sell you stuff.
Upgrade weapons and increase base stats
Weapons are organized in rarity, as are armor and other pieces of equipment. Keep the rarest of the rare, and then use the other goodies as fodder so that you can enhance your equipment. When you do, the equipment will level up and all of its stats will be boosted. Once it reaches max level, and you have the right materials to do so, you can then "tier up" a weapon to increase the base stats permanently.