Prototype

GLITCH WARNING: It is possible due to a glitch to miss out on the Pt trophy, but thankfully it's easy to avoid. The glitch occurs if you obtain Platinum Medal scores in all of the events before you unlock the Au trophy. To avoid this and make sure you can get Pt, simply make sure that you have at least one event that you have not obtained a platinum level score on before you unlock Au. As long as you have at least one event that stays at Gold when the game gives you the option to go for platinum scores, you will not have any problems. Thanks to blymberry for discovering this and bringing it to my attention.

Before the game will show you Platinum Medal times and scores for events, you will have to achieve Gold Medals in all of them first. However, any scores you have that already qualify for a Platinum Medal will be upgraded once the scores are revealed. You will unlock events as you play through the story, and once you have finished it they will all be available to play. ThisIsWar was kind enough to assemble a collection of videos that show how to complete these events to the Platinum Medal standard better than I could ever explain in words, so please see the following posts if you are having trouble with an event.

Movement Events: Movement Events - View Single Post
Glide Events: Glide Events - View Single Post
Kill Events: Kill Events - View Single Post
War Events: War Events - View Single Post
There are also some good tips pertaining to some of the events in this thread which I also recommend that you look over if you are having trouble: pt trophy advice given here, ask any questions you need - PS3Trophies.org

ChronoAlien provided me with these tips on using Alex's Airdash ability, which proved to be very important information I needed to complete some of the tougher movement events:
"Doubledash is really what is says, but it's all about the angle when you use them. Alex will dash the same way he is traveling, so naturally, if you wanna move fast across roofs, you want to make sure he's going straight before you dash. If he isn't, you will dash either upwards or downwards, wasting valuable time. If I wanna travel fast, I dash fast, and I NEVER use the Glide ability on 'movement events.'
Climbing Doubledash is just a little advancement really. 'Point to Point' event is a great place to master it, as the second mark, and last mark requires you to use them, almost at least.
As soon as Alex leaves the ground using jump, as he is traveling upwards, doubledash to make him continue traveling, hence, making him climb with a doubledash. This saves you a full second, even 2, instead of having to dash into a wall and then running up."

You can track your progress under the Events page on the pause menu (there is a separate tab for each kind of event).