Page 116 of the 5e PHB defines divination as "seeking a clearer understanding of the past, present, and future". They strive to "part the veils of space, time, and consciousness to that you can see clearly." That certainly fits the general view of divination, but what does it really mean and how do you include that in a game like D&D?
The 5e system reduces divination into two things: ways to affect game mechanics (substitute dice rolls), and sensory effects/skrying. Here are all of the spells in the School of Divination:
Guidance Cantrip
True Strike Cantrip
Beast Bond 1st
Comprehend Languages 1st (rit.)
Detect Evil and Good 1st
Detect Magic 1st
Detect Poison and Disease 1st (rit.)
Hunter's Mark 1st
Identify 1st (rit.)
Speak with Animals 1st (rit.)
Augury 2nd (rit.)
Beast Sense 2nd (rit.)
Detect Thoughts 2nd
Find Traps 2nd
Locate Animals or Plants 2nd (rit.)
Locate Object 2nd
Mind Spike 2nd
See Invisibility 2nd
Clairvoyance 3rd
Tongues 3rd
Arcane Eye 4th
Divination 4th (rit.)
Locate Creature 4th
Commune 5th (rit.)
Commune with Nature 5th (rit.)
Contact Other Plane 5th (rit.)
Legend Lore 5th
Rary's Telepathic Bond 5th (rit.)
Scrying 5th
Find the Path 6th
True Seeing 6th
Foresight 9th
Only the spells highlighted really have anything to do with potentially seeing into the future. Even Legend Lore is focused on the past.
In the Nolaria game system, Divination is a collection of spells, objects, skills, and other things that convey knowledge or information about events that happen in the past, present, or future. Events in the past have to do with lore, hence Legend Lore. Current events, including sensory ones generally fall into the category of skrying. Knowledge of future events is called forecasting or foresight.
Classically, there are skills, objects, etc. that can enhance the ability of a diviner to get a better understanding of the past, present, and future. In particular, there is a body of skills based on seemingly random things that enable a talented individual to improve their sight. These methods include:
Astrology
Casting of Bones
Ceramancy
Divining Rods and Wands
Enchanted Dice
Graphomancy
Ink Splatters
Palmistry
Reading Entrails
Reading Tea Leaves
Tarot - by extension
Ur Principle
Water Bowl Divination
Yarrow Sticks
These techniques have affinity for certain areas of knowledge. For example, Ink Splatters are used to remember things, to explore things hidden inside of a person. It is associated with mental states, attitudes, and memory.
Rather than relying on a pure game mechanic, the results are crafted by the DM to further a story line, provide richer detail, insight into motivations, etc. Gaining these skills happens over time and requires access to teachers, mentors, equipment, library material, and other things.