FATEs are located in several areas, and show up on your mini map as a purple icon with a blue circle around them. When you enter the blue circle, you join the FATE and help everyone to achieve a common goal. Stay inside the blue circle until the FATE ends and you will get credit towards this.
Please note, depending on your level vs the level of the FATE you may see an option pop up to the right of your screen that asks you to Level Sync. Be sure to click on that option so the FATE will count by automatically adjusting your level as needed.
You can also get excellent experience by joining FATE groups. You may see people advertising in chat for "LFM FATE Group!" if you send them a /tell they will invite you and you can follow the group from FATE to FATE racking up combo kills and big bonus exp rewards. This helps the grind to 1000 fates go by much faster and levels you up in the process.
Dont forget to eat or drink a buff item that not only increases stats but gives bonus exp, usually for 30 minutes. You can buy stacks of 99 of these from vendors in town. Look for whatever food or drink compliments your primary stats for best results.
Always be sure to log out in a sanctuary area to acquire rested exp while you are offline. These are normally found in the main cities by the large teleport crystals or by purchasing a room at an inn. you will know you are in one when you see a small crescent moon above your experience bar at the bottom right of the screen.
To check your progress towards this, press Start then scroll down to the Achievements menu. Select that, then select the Battle tab. Scroll down until you see Date with Destiny IV. Select that, and you will see how many you have completed so far.