Level ups happen much like they do in most games these day. You gain experience from quests or killing creatures, and you gain level ups.
Upon leveling, a player will receive 3 points to add to their persona's stat points distributed between Strength, Magic, Endurance, Agility, and Luck. No more than 2 points can be put towards a stat per level up, thus you have to put either +1 to three different stats, or +2 to one stat and +1 to another. The next level up, if you put +2 to a stat, you can't put anything into it. Then the level up after that one you may only put +1 in that stat, then after that you may put another +2 in it, which repeats the process. In the end it looks like Strength gets +2, +0, +1, +2, +0. +1, +2. Thus in 7 level ups, your persona gained 8 points in that one stat. However, doing +1 has no regulations, thus in 7 levels, you have 3 stats with 7 points in them each or 2 stats with 8 points each. But remember, there are 5 stats, all of which are important; don't focus down only 2 or 3 stats.
It should be noted that a player will gain a certain amount of HP and SP upon a level up as well.
Persona can evolve under certain circumstances. There are two types of evolution, and specific actions must occur before either can occur;
Metamorphosis: When something drastically happens to change a person's outlook upon life, or something drastic happens to them, their arcana may possibly change to represent this and thus their persona will change to more properly fit their new arcana, their new outlook, and new personality. Although it doesn't happen often, it can happen and with terrifying results in most cases. Note that the only thing that happens is the persona becomes a different persona altogether with different weaknesses and strengths, and skills that more accurately portray it. No bonus stats are gained.
Evolution: This is a straight evolution of the persona. The persona evolves into a stronger version of itself, becoming a new persona altogether. It should be noted that the arcana does not change in this case, but rather becomes it's truest form. A player must reach level 50+ before this can occur. Upon occurring, their level is reset to 1. The stats stay the same as they were before the reset to level 1. However, the persona does gain bonus points to add to their stats. For an evolution to occur, besides having to reach level 50+, a quest must be done to uncover your true self. Examples of this can be seen amongst the allies of the Protagonist from Persona 3 and Persona 4; IE Yosuke's Persona, Jiraiya becomes Susano-o when he finally begins to be whom he truly is.
There is a max of 99 levels in this game. Depending on how long the game lasts, the cap may be increased. However, the only way to get more than 99 levels is through evolution. Upon evolving, levels are reset to 1, and the persona can gain stats from 1-99 again. It is possible, and recommended for your sake, to level up to 99 before attempting an evolution quest. This makes it where, when you reach 99 in your evolved state, you will be an accumulated 198 levels. It should be noted that this doesn't affect when you gain abilities. If you were level 50, and your next skill to be learn is at 52 but you evolve, your skill requirement levels will scale downwards making the skill you'd have learned at 52 be one you'll learn at level 3 instead (2 levels later, just as before). This isn't really an advantage, as it's the same number of levels. Experience stays the same upon evolution. If, for example, you needed to gain 90,000 experience to level up to level 51 from level 50, upon evolving at level 50, the experience to level up from level 1 to 2 would remain 90,000. Thus the advantage in evolution isn't in learning skills earlier or less experience to gain, as the experience necessary to level up would be the same had you not evolved and the skills are the same levels away as before. The advantage is both in the stat bonus from evolving, and the number of extra levels. Remember, there is a max of 99 levels. Getting as many levels as possible before evolving means more level ups in the end. Getting from 1-50, then evolving makes you only get another 99 means only 149 levels total. Getting from 1-99, then evolving and getting another 99 means 198 levels total. Reflect upon these choices wisely, as an evolution bonus is very helpful, but it'll be short lived if you did it prematurely and other players gain more levels. The reason this was stressed upon so much is to make sure you realize when it is possible, what the benefits are, and when you should properly do it; all to help you play better. As stated before, level cap may be increased in the future, but most likely only for evolved states and not the unevolved persona.