Early on it's a sprint to reach level 85 faster than your opponents in arenas. Doing all daily challenges gives you extra bonus experience points to level up faster.
Buying energy refreshes to burn more energy and level up faster.
In general until you're sure to have a large surplus of credits, only level up characters, that you're sure you will use/need in the near future. Always keep your arena team on max. level.
Phoenix are still often the first team to focus on. It benefits from also being rebels (they unlock EP as well as Thrawn) and two of them are jedi as well (used to unlock Grand Master Yoda).
The drop rate for character shards is 33%. There's nothing you can do to change that. If you want to farm them faster, buy more energy refreshes.