and how do you make sure you've done it well enough that later on, the characters voice and action don't seem like a totally different person. i'm sure that writing he or she said this simply won't work. and how do you make sure they stay within character? Not sure if that makes sense... what i'm trying to say is, making your character act and talk the way you envision him to, personality wise..
Subtlety is the best way. Change the way they think about similar situations from past to present, like maybe a political belief they may be strong about where they change their opinion slightly due to some events. Just keep easing them into another mindset throughout the story, so that everyone can see the transformation.
That done more through actions rather than speech mostly. Your character begins to be more conservative in his/her actions--you show them being more cautious in their movements; show them anticipating the antagonists actions.
It's a hard thing to do as it can be done in many different ways...someone can't give you instructions how to do it, it won't work. Reading is a good thing to do for these kind of things. So pick up some good books and get reading
It's all in transitions. If you want your character to change between the beginning and an end, that's great because a character arc is essential for any good story. But think about yourself, how do you change? Either, there's a massive event that freaks you out on some level and you instantly change something that you realised was hurting you (say, a smoker leaves cigarettes after a heart attack or someone weighs themselves, realises they gained twenty pounds over the winter, and in reaction to that they stop eating desserts and join a gym straight away). But then you still have to go through a period of getting used to the new you, and if it's done so abruptly, there are cravings for the old behaviour, whatever it is, smoking, cheating on a spouse, going out with gangsters, junk food etc. So the bulk of the story is in this 'getting used to it' phase. Or you can engineer a series of events which are significant enough to make your character realise things bit by bit, and change slowly, as the story develops. In both cases, the bulk of your novel will be focused on reacting to stimuli and gradually changing from within. I think those are two basic approaches, but you have to find a way to show it in your book effectively. A character arc usually lasts the entire book, so you really need to take your time exploring it in a meaningful way. Get down to the nitty-gritty, that always strikes home for the readers
A common technique used to do what you want to do is putting the character in similar situations and showing how he reacts differently. Other ways you can do it is by showing how he treats people differently, how his behavior or his thoughts are different from his past self etc. Ultimately you will have to think of it yourself, since that is the job of the author.
Generally, my works are based upon quite long courses of time during which characters are changing. To ensure, they are still the same people I try to emphasize their specific deeds typical only of them.
Know your characters intimately. You're the omniscient god of the story world they live in. Their personalities, values, etc. are products of their past experiences and continue to develop with the obstacles you place in their path. It's all about their ability to shift gears and overcome the hardships that are thrown at them. When a character is faced with a situation, you have to ask yourself, "Would he/she really do that?" Like Youniquee said, the best way to learn is to read. Look back at your favorite books and study how the authors use literary tools (action, dialogue, interior monologue, etc.) to bring their characters to life and show how they adapt and grow as the story progresses, while still keeping them in character.