There are a number of trainers for the various skills in Skyrim. All skills can be improved to a maximum of 100 points, but trainers will only train the Dragonborn up to 90 points, the rest must be "earned" through practice.
Trainers exist for these skills:
Alchemy • Alteration • Archery • Blocking • Conjuration • Destruction • Enchanting • Heavy Armor • Illusion • Light Armor • Lockpicking • One-handed • Pickpocketing • Restoration • Smithing • Sneak • Speech • Two-handed
Milore Ienth
Alchemy
75
Expert
Solstheim , Ienth Farm
Dragonborn DLC
Lami
Alchemy
50
Common
Morthal
Arcadia
Alchemy
75
Expert
Whiterun , in Arcadia's Cauldron
Babette
Alchemy
90
Master
Dark Brotherhood Sanctuary
Dravynea the Stoneweaver
Alteration
75
Expert
Kynesgrove
Tolfdir
Alteration
90
Master
College of Winterhold
Faendal
Archery
50
Common
Riverwood
Aela the Huntress
Archery
75
Expert
Whiterun , in Jorrvaskr
Niruin
Archery
90
Master
Riften , Thieves' Guild
Sorine Jurard
Archery
90
Master
Fort Dawnguard
Dawnguard DLC
Only if you join the Dawnguard
Njada Stonearm
Block
75
Expert
Whiterun , in Jorrvaskr
Chief Larak
Block
90
Master
Mor Khazgur
Runil
Conjuration
50
Common
Falkreath , in the Hall of the Dead
Talvas Fathryon
Conjuration
90
Master
Solstheim , Tel Mithryn
Dragonborn DLC
Phinis Gestor
Conjuration
75
Expert
College of Winterhold
Falion
Conjuration
90
Master
Morthal
Wuunferth the Unliving
Destruction
50
Common
Windhelm , in the Palace of the Kings
Sybille Stentor
Destruction
75
Expert
Solitude , in the Blue Palace
Faralda
Destruction
90
Master
College of Winterhold
Neloth
Enchanting
75
Master
Solstheim , Tel Mithryn
Dragonborn DLC
Sergius Turrianus
Enchanting
75
Expert
College of Winterhold
Hamal
Enchanting
90
Master
Markarth , in the Temple of Dibella
Only after completing the quest The Heart of Dibella
Kuvar
Heavy Armor
75
Master
Solstheim , Thirsk Mead Hall , Bujold's Retreat
Dragonborn DLC
only after Retaking Thirsk , lie about Bujold the Unworthy right to lead.
Isran
Heavy Armor
90
Master
Fort Dawnguard
Dawnguard DLC
Only if you join the Dawnguard
Gharol
Heavy Armor
75
Expert
Dushnikh Yal
Farkas
Heavy Armor
90
Master
Whiterun , in Jorrvaskr
Only once the quest Glory of the Dead has begun, and thereafter
Atub
Illusion
75
Expert
Largashbur
Drevis Neloren
Illusion
90
Master
College of Winterhold
Scouts-Many-Marshes
Light Armor
50
Common
Windhelm
Grelka
Light Armor
75
Expert
Riften , near the marketplace or at Haelga's Bunkhouse
Nazir
Light Armor
90
Master
Dark Brotherhood Sanctuary
Ma'jhad
Lockpicking
75
Expert
Khajiit Caravan either near Solitude or Windhelm
Ra'zhinda
Lockpicking
75
Expert
Khajiit Caravan either near Solitude or Windhelm
Vex
Lockpicking
90
Master
Riften , The Ragged Flagon
Amren
One-Handed
50
Common
Whiterun
Athis
One-Handed
75
Expert
Whiterun , in Jorrvaskr
Burguk
One-Handed
90
Master
Dushnikh Yal
Silda the Unseen
Pickpocket
75
Expert
Windhelm , outside the Palace of the Kings
Vipir the Fleet
Pickpocket
90
Master
Riften , Thieves' Guild
Florentius Baenius
Restoration
90
Master
Fort Dawnguard
Dawnguard DLC
Only if you join the Dawnguard
Colette Marence
Restoration
75
Expert
College of Winterhold
Keeper Carcette
Restoration
75
Expert
Hall of the Vigilant
Danica Pure-Spring
Restoration
90
Master
Whiterun , in the Temple of Kynareth
Only after completing the quest The Blessings of Nature
Aphia Velothi
Restoration
50
Common
Solstheim , Caerellius House
Dragonborn DLC
Ghorza gra-Bagol
Smithing
50
Common
Markarth , at her forge
Balimund
Smithing
75
Expert
Riften , near the marketplace
Eorlund Gray-Mane
Smithing
90
Master
Whiterun , at the Skyforge
Only if you join the Companions
Gunmar
Smithing
90
Master
Fort Dawnguard
Dawnguard DLC
Only if you join the Dawnguard
Khayla
Sneak
50
Common
Khajiit Caravan near Whiterun
Garvey
Sneak
75
Expert
Markarth
Delvin Mallory
Sneak
90
Master
Riften , The Ragged Flagon
Dro'marash
Speech
50
Common
Khajiit Caravan near Dawnstar or Riften
Revyn Sadri
Speech
50
Common
Windhelm , in Sadri's Used Wares
Ogmund the Skald
Speech
75
Expert
Markarth
Giraud Gemane
Speech
90
Master
Solitude , at the Bards College
Ronthil
Speech
50
Common
Castle Volkihar
Dawnguard DLC
Only if you join the Volkihar Vampire Clan
Fura Bloodmouth
Two-Handed
75
Expert
Castle Volkihar
Dawnguard DLC
Only if you join the Volkihar Vampire Clan
Torbjorn Shatter-Shield
Two-Handed
75
Expert
Windhelm , in Candlehearth Hall
Vilkas
Two-Handed
90
Master
Whiterun , in Jorrvaskr
Wulf Wild-Blood
Two-Handed
90
Master
Solstheim , Skaal Village
Dragonborn DLC