How to quickly gain experience in Torment: Tides of Numenera?

Interactions with the environment will provide you with additional experience points

Experience is necessary for reaching new tiers and levels of knowledge. Each new level requires more experience to gain. Experience is given to you for:

• Completing quests - It is the most profitable and fastest way for gaining new levels. By completing quests you receive a large number of experience points that can be used on developing characters. It is worth adding that in most cases you are awarded only for completing the quest - combat and interactions during the quest usually aren't awarded with experience. In other words, fastest ways of completing quests will be awarded in same way as the longer ones (at least when it comes to experience; items or character attitude may vary).
• Exploring the world and interacting with it - In the game world you will find plenty of interesting characters, items or other pieces of the world you can interact with. Many of them offer a chance to complete a short test as part of the effort mechanics. For example, you can try fishing a staff from the water. These actions provide additional experience points (usually 2-3 EXP) and they enable you to obtain new items. You should click on each interesting item so that you won't miss anything.
• Use the Anamnesis skill - Anamnesis is useful during the whole game and it enables the main character to remember distant memories. Usually it is awarded with additional experience points.

In general - the fastest way of reaching new levels is by focusing on the story and completing quests. Additional experience points can be gained by interacting with the environment and by using Anamnesis.