Some clarification first.
Does the Pact of the Tome Warlocks learn new ritual spells from other classes?
(PHB 108) When you gain this feature, choose three cantrips from any class’s
spell list. While the book is on your person, you can cast those
cantrips at will.
You can only get 3 cantrips as stated. Another reason why it does not allow you to research new spells is because of the Incantation "Book of Ancient Secrets", and that send us to the following point.
Book of Ancient Secrets
From the description (PHB 110):
You can now inscribe magical rituals in your Book of Shadows.
With your Book of Shadows in hand, you can cast the chosen spells as rituals.
On your adventures, you can add other ritual spells to your Book of
Shadows. When you find such a spell, you can add it to the book if the
spell’s level is equal to or less than half your warlock level
(rounded up) and if you can spare the time to transcribe the spell.
With "Book of Ancient Secrets" invocation you can learn new ritual spells. These spells can only be cast as rituals.
There are two meanings to the word "research" referent to wizards and spells; the homebrew type of research, where it is implied that you create new and custom spells, and the to add new (but existing in the game) spells to the wizard spell book.
This type of "research" allows to create custom spells, as in literally new spells and modifications of existing spells. In the first instance you create spells based on an idea or an existing spell from other editions or games. In the second instance you take one spell, let say magic missile, and modified it to be more fun or align with a thematic; e.g. "Elemental magic missile". There exist guidelines in the DMG for homebrewing spells and has to be worked with your DM.
The way you create custom spells vary from table to table, some might just add to the wizard spell list, other would ask you to use your downtime time to "research" the spell and add to the list, or it can be the reason for an entire adventure. On the last point, just to add a bit of history, there are prominent wizards (that might or might not be relevant or exist in your game) through the history that left their work for future generation of wizards, two of the most common examples are Mordenkainen and his disciple Bigby.
From PHB 114, emphasis mine.
Copying a spell into your spellbook involves reproducing the basic
form of the spell, then deciphering the unique system of notation used
by the wizard who wrote it. You must practice the spell until you
understand the sounds or gestures required, then transcribe it into
your spellbook using your own notation.
The cost represents material components you expend as you experiment
with the spell to master it, as well as the fine inks you need to
This segment feels like a typical day for a researcher. You pick an article (and if interest you), some times you need to decipher the notation, you keep reading and reading until you are sure that you understand the article, and then you transcribe it into your own article (or thesis). If in doubt of the results, you do the often undervalued action of trying to reproduce the experiment.
In PHB page 187 it is describe the action of researching as a downtime activity, emphasis mine. This can be ruled as a way to create homebrew spells as a mean of "research", or use the downtime to find spells and transcribe them in you spell book. This is up to the DM, though.
The time between adventures is a great chance to perform research,
gaining insight into mysteries that have unfurled over the course of
the campaign. Research can include poring over dusty tomes and
crumbling scrolls in a library or buying drinks for the locals to pry
rumors and gossip from their lips.
Note: Store playing
Bear in mind that store playing, as in Adventure League, might be against of homebrewing since they try to keep thing as balance and uniform as possible so you can move your character from table to table.