There are two methods to building characters: Slowly or All At Once.

Slowly. You start with a basic idea of the role you need your character to fill. You start shaping the most basic personality around it. For example, you know a character will become the rebel leader, so you make them rebellious and a leader. You might make vague efforts to describe them, like brown hair and pudgy. Then you start writing the story. Your amorphous character begins to encounter conflict and times for development. You make decisions on how a character will react to that conflict, to that time of development, and your character becomes the aggregate of those decisions and developments. 

All At Once. You fill out character questionnaires, make or find a body/face that fits them, and map out their growth. My favorite questionnaires are here and here. You can also use the immensely useful Charahub to keep track of minor characters you don’t want to fill out a full questionnaire for.

For more help on either method, look in my character building tag and personalities tag, and there are more specific tags under the character heading.

What you change is your decision. All characters should undergo some kind of change. Most major character arcs go from worse to better, although some go from better to worse. How it changes is also your choice. How someone goes from being a nobody to a billionaire will be different than someone whose pride degenerates into arrogance.

The key to change is something you’ve already mentioned. Take it slow. Here’s how I go about character change:

• Starting point. The character has not undergone change. Introduce how they are before their developing arc. 
• Vague movement in direction of change. The character moves/is nudged in the direction of change. A bourgeoisie worker recognizes one of merit of communism while despising the rest.
• More vague movement. The character makes a more definite action towards change. A character checks into the hospital to visit their estranged mother before chickening out.
• Quasi-commitment. This is the halfway point, where the character arc is slowly moving towards its conclusion. A character who is always honest sees how their honesty might hurt people/themselves and begins to tell a few lies.
• Still has reservations. The character is mostly devoted to their change but hangs on to their old ways. They may be offered a chance to go back to their old ways, but resist (or don’t). A heroic character must choose between saving the orphanage or giving his soldiers a desperately needed rest. He chooses to give his soldiers a rest while feeling twinges of guilt over the dead orphans.
• Transformation. The character has completed their arc. *throws confetti*

There isn’t a set time on how fast these events take place. They could take place over weeks or years. They could even take place over days if the character is under a lot of pressure and/or not thinking rationally. Your character may be aware of the change they’re going through. Generally, the shorter the time span, the more aware they will be of the change. Even if they’re not aware of the change, their close companions (friends, family, lovers, etc.) will be. They may call the character out on their change or encourage them to embrace it. Or they could do nothing.

One of the easiest ways to show at what point of development the arc is in is to give the character a chance to return to start. If they’re early in development, they are more likely to accept. If they’re late in development, they are more likely to refuse. Characters at this point rarely accept reversion, although a character arc that rises to a point before descending to previous levels is also acceptable. (Many authors use this technique to make their heroes darker: the hero makes increasingly darker choices before their saintly friends/love interest pulls them back into the light.) A character may accept reversion before deciding the old ways aren’t them anymore and continue with their change.

You can read more about pacing character change here.