According to most psychologists, our personalities are sort of like teeth: they're stable from day to day, but can shift over time. So how do we shift them? With teeth, it's clear — orthodontia! — but with our personalities, it's fuzzy. Is your personality just an accumulation of life experience, or can you change your personality on purpose?
A Personality Workout
A recent study explored this question by having people perform workout regimens, more or less, for their personalities. To start, researchers recruited 377 psychology students and had them take a 60-question personality test that measured the "Big Five" personality traits: openness, conscientiousness, extraversion, agreeableness, and neuroticism.

Next, they asked the participants if there were any traits they wanted to work toward changing. Typically, they picked two traits they wanted to work on; the most popular goal was reducing neuroticism (unsurprising in our age of anxiety and anxiety baking).
Every week for the next 15 weeks, participants accepted up to four "challenges" designed to push them towards their goals. These were designed to be "small, reasonable and measurable" ways to change thought patterns and habits — the type of concrete tips you might see in a self-help book.

For instance, a challenge to prevent neuroticism was, "When you feel worried about the future, spend at least two minutes visualizing the best-case scenario." People striving for more extraversion, meanwhile, might tackle challenges like "Introduce yourself to someone new." Just like a workout regimen, over time, people's suggested challenges got harder.

At the end of each week, the participants reported whether they had completed the challenges they accepted and also retook the personality test. By the end of the study, a lot of them had made progress towards their goals of changing their personalities — but there was a catch.

For the most part, researchers found that the more "challenges" a person completed, the more progress they made. Aspiring extroverts got more extroverted if they completed 30 challenges in 15 weeks than they did if they completed 20. The same held for people working on their conscientiousness and neuroticism. (Openness and agreeableness, however, were more stubborn and relatively unaffected by challenges.)

Still — so far, so intuitive. The more steadily people worked towards their goals, the more they succeeded.
Here's what's weird, though. Signing up for a challenge and not completing it seems innocuous enough — no progress, no loss, either. In reality, though, it correlated with backsliding. A person working to be more extroverted, for instance, often became less extroverted when they couldn't complete a challenge. Follow-through, in this study, was more important than anyone expected.
Researchers aren't exactly sure why this is. Since the study relied on participant-reported personality data, it may have just been a sign that failing a challenge made people perceive their personality "issue" as bigger than before. Then again, maybe the very act of accepting a challenge felt like progress and made them relax more deeply into the trait they wanted to fix — the same way you might overeat after a week of successful dieting.

Overall, the study seems to confirm that you can, indeed, change your personality, but you can't just want to change it. You have to work at it steadily and concretely. It also seems you have to work at changes like this for a long time, possibly forever. This study only lasted for 15 weeks, and it's unclear if the progress participants made continued beyond that.
So before you take on a project like becoming more extroverted, make sure you're seriously committed to it — or you could end up more introverted instead.