The first part of the game to be unlock are the Light Side Battles. At this point you should have a squad of three characters:
Jedi Consular
Clone Sergeant - Phase 1
Clone Wars Chewbacca
The game is still walking you through itself, so just follow the onscreen instructions to get to your first proper battle.
While in battle you can tap and hold each of the action buttons, this will bring up a description of each move. Take note also of the cooldown period. For example the Jedi Healing is reusable in 5 turns at this stage of the game, so time your usage of your healing moves well.
The initial battles are really, don't forget you can also tap on one of the enemies to select them to be attacked, a target icon will appear around their head when they are selected.
At the end of each battle you are given a series of rewards, XP, credits, gems, Training Droids shards, sim cards, upgrade chips, etc... It can be any combination of the above.
You can use these to upgrade your characters and unlock new ones.
If one of your squad members have a special leader's ability, then they should be placed in the first slot when you pick your team. At the beginning of the game, your Wookie is the only team member with a leader's ability (Wokkie's Resolve), so he should always occupy the first slot.
When you are in a battle there are two bars, a green one and a blue one. The green bar reflects the character's health, this will drop as he takes hits and turn yellow, the red until finally you have energy left.
A good time to use your healer's ability would usually be when members of your team have gone into the yellow zone. As the game progresses and you become used to your characters you'll get a better feel for the best time to make energy boost moves.
Below the energy bar is the speed bar, this indicated which character will move next. Once the blue bar is filled, the character will make his move, followed by the next one to get a full blue bar and so on. Keeping an eye on the blue speed bar will allow you to make decisions based on whose turn it is next and whether or not you have to use a heal for example, or if you have time for your tank to make a taunt and take the pressure off the rest of the team.
Sometimes during a battle a character will have a status effect applied against them. Red ones are negative effects, and green ones are positive effects. You can press and hold the effect icon for details.