Protoss vs Protoss won't be much of a problem if you know which unit counters which. It is also important to gain an upper hand in reconnaissance, time, and economy. Gaining this advantages will mean you the victory you want to achieve and prevent war of attritions which is a common scene during these match ups. How can this be achieved and dominate a Protoss vs Protoss game?
A great way to open up a good offensive is by rushing in Zealots. This is a common practice for Protoss players to harass miners and cripple economies. If left unguarded, a Zealot rush against a Protoss could mean victory. In an event that your opponent uses the same tactic, it is imperative to use micromanage your units to effectively kill off their Zealots. Your opponent will grieve at the sight of you dominating his Zealots. You can then try to penetrate his base and kill off his miners.
In case a Zealot rush proved to be ineffective, a good intel is essential to win a Protoss vs Protoss game. Knowing your opponent's moves will allow you to react accordingly, depending on what your opponent plans. Look for their current buildings, the units they can muster, structures on cue, the works. This will give you ample information to what your enemy is capable of doing for that moment.
It is also essential to know the counters for each Protoss unit to move according to the situation. Knowing this vital information will allow you to effectively thwart almost anything your opponent throws at you. If your opponent is focusing on creating Colossi, build many Immortals. Immortals boast energy shields that can tank the Colossi's anti-personnel lasers. As the Colossi are busy with your Immortals, you can then finish off the giants with Dark Templars or Zealots.
If he is focusing on creating a fleet, mass up Stalkers. Stalkers are the Protoss best weapon to counter air units and are capable of surprise attacks using their blink skills. Countering their army with the right units will get you through the game with little casualties and massive losses for your enemy.
Another thing to remember is that it's always a good move to destroy enemy expansions. The primary purpose of an expansion base is to gather-up resources to fund their little schemes. Prevent them from gaining this advantage by scouting expansion areas with observers and sniff out all their expansion bases. This will force your enemy to desperately fortify expansion bases thus funneling resources to defend the colony. Exploit the situation by constantly harassing the expansion with Phoenixes at the same time amass a decent attack force at their main base. This will create confusion and eventually wear down their forces. You can then sweep in with Colossi, Templars, or Void Rays to end his duffering.
A fight between Protoss and another Protoss will require you to use micromanagement skills extensively. The key is to gain an upper hand in terms of economy and speed and to keep a constant intel on your enemy movements. This strategy will allow you to counter every move your opponent makes and ultimately result to an undisputed victory.
[Victory Goes to the Swift]
With a Protoss versus another Protoss battle, victory goes to the swift. Your ability to counter attacks your opponent dishes out on you will be tested here. It is important to anticipate and scout as much as you can. Try to anticipate what units, whether aerial or land-based units the enemy will be using so you can craft-up units to counter it. Void Rays will be dealt easily with Stalkers in the same way as Stalkers and Zealots are dealt with effectively by using Colossus while using your own zealots to defend it from any direct attacks it can receive. For those stealthy Dark Templars, make sure you have your Observers patrolled at strategic points in your base. Defending your base actually is easy if you know what units to use to counter enemy attacks and victory for you is certain if you know what consequences or conditions the enemy gets in producing or using a certain unit in the battlefield like expensive Carriers will surely mean less defensive abilities on the enemy's base as most of his resources are used up in creating his Carriers. The tactics mentioned are just but a few of the many ways to finish off the same Protoss race. Just remember, be quick, anticipate what units the enemy will use and be able to counter it.