It is easy to notice that there are certain quests that provide an increase in skill level however after a bit of thorough Googling there doesn't seem to be a proper list of them. I concluded that not very many people who play this would even bother to edit a wiki for this topic.( as most of them would probably be too busy playing Skyrim)

So i have decided to compile a list of what i found so far and urge anyone who sees missing information to provide input by replying to this topic:

Thief Play-style:

• Retreive Finn's Lute for Inge Six Fingers at the Bard's College in Solitude( increases all following Theif skills )

Light Armor

• Defeat bandit leader for Annekke in Darkwater Crossing
  
• Retrieve Amulet for Shahvee in Windhelm argonian assemblage (note: also increases lockpicking)

Sneak

• Return helmet to Noster Eagle-Eye in Solitude
  
• Retreive bottle of Double Distilled Skooma for Stands in Shallows in the Argonian Assemblage in Windhelm

Lockpicking

• Retrieve Amulet for Shahvee in Windhelm argonian assemblage (note: also increases Light Armor)

Pickpocket

• nothing yet

Speech

• Steal the Staff of Arcane Authority for Malur Seloth in Winterhold.
  
• Collect a Mammoth Tusk for Ysolda from Whiterun.
  
• Jumping from Bard's Leap Summit at Lost Valley Redoubt south of Roriksted and surviving will raise your Speech when you talk to the Bard's Ghost at the bottom.

Alchemy

• Retrieve Ring of Pure Mixtures for Frida in Dawnstar
  
• Find Song of the Alchemists for Lami in Morthal.

Mage Play-style:

• Find Pantea's Flute for Pantea Ateia in the Bards College in Solitude( increases all following mage skills)

Illusion

• Defeat the Master Vampitre for Sybille Stentor in Solitude

Conjuration

• nothing yet

Destruction

• nothing yet

Restoration

• nothing yet

Alteration

• Retreive Frost salts for Dravynea the Stoneweaver in Kynesgrove

Enchanting

• nothing yet

Warrior Play-style:

• Retrieve Rjorn's Drum for Giraud Gemane in The Bard College in Solitude( increases all following warrior skills )

Smithing

• Retreive Queen Freydis' sword for Oengul War-Anvil in Windhelm
  
• Retreive Night Falls on Sentinel for Ruslief in Dawnstar
  
• Retreive The Last Scabard of Akrash for Ghorza gra-Bagol in Markarth

Heavy Armor

• Kill bandit leader for Brunwulf Free-Winter in Windhelm

Block

• Retreive Amren's family sword for Amren in Whiterun (note:completing this also increases one-handed skill)
  
• Retreive Shield for Roggi Knot-Beard in Kynesgrove
  
• Retreive Hjalti's sword for The Ghost of Old Hroldan at Old Hroldan Inn (note: also increases one-handed skill)

Two-Handed

• Solve Octieve San's debt problem in Solitude

One-Handed

• Retreive Amren's family sword for Amren in Whiterun (note:completing this also increases Block skill)
  
• Retreive Hjalti's sword for The Ghost of Old Hroldan at Old Hroldan Inn (note: also increases Block skill)

Archery

• Practice with Angi at Angi's camp (note: increases archery by 6 points)

Completion of Discerning the Transmundane for Septimus Signus in Septimus Signus's outpost (North north-west of College of Winterhold) provides item that can increase your choice of Theif, Mage or Warrior skills by +5

Archery - Agni near Falkwreath. The quest is literally training alongside her in archery.

thank you for your contribution. Your input is very helpful and i have added it to the list

Spoiler section please.

Though here's some you can add.

i know those ones but they increase sneak and illusion (Noster Eagle-eye and Sybille Stentor)
but if you do come across destruction increase or any of the other missing skill increasing people please share

Oh yeah you're right about the old guy, but the mage gave me "D" and not "I".
(I just did the quest before posting).

Edit : btw i'm certain, just tabbed in and out, i have 15 "I" and 16 "D"

One of the daedric quests gives you a book that increases all skills for the category you choose - Combat, Magic, or Stealth.

thanks for reminding me i totally forgot to add that one.....might as well add it even though it hasn't got a specific category
well actually I probably didn't add it because it involves a book however it does require a quest to get so it deserves to be up there anyway.

what is the name of the npc? because i tried Sybille Stentor again and she gave me illusion

*bump* need more info please