Trainers who sell as well as train

The full list of trainers capable of serving as merchants is as follows:

Base Game

• All but one of the trainers at the College in Winterhold also sell various magical supplies; typically books related to their school, robes, and soul gems.
  • Tolfdir (Alteration)
  • Colette Marence (Restoration)
  • Drevis Neloren (Illusion)
  • Faralda (Destruction)
  • Phinis Gestor (Conjuration)
  • The one exception is Sergius Turrianus, who doesn't sell anything, despite even having the Spell Vendor class.
• In addition, several of the non-College-affiliated, magic-related trainers are also merchants.
  • Conjuration has Falion.
  • Destruction has Sybille Stentor and Wuunferth the unloving.
  • Illusion should have Atub, but unless you have the Unofficial Skyrim Patch, she is bugged, and not properly added to the factions that would enable this.
• All three Alchemy trainers are also alchemy merchants.
  • Lami in Morthal
  • Arcadia in Whiterun
  • Babette in the Dark Brotherhood Sanctuary
• Similar to Alchemy, all three Smithing trainers are also blacksmithing merchants:
  • Ghorza gra-Bagol in Markarth
  • Balimund in Riften
  • Eorlund Gray-Mane
• There are a few assorted NPCs who are both trainers and merchants outside the above.
  • Gharol in Dushnikh Yal is a Heavy Armor trainer, and also a blacksmithing merchant.
  • Grelka in Riften is a Light Armor trainer, and also a merchant of various items, mostly blacksmithing-related.
  • Ma'jhad, of one of the khajiit trade caravans (going between Windhelm and Solitude), is a Lockpicking trainer, and also a fence with the thieves' Guild.
  • Revyn Sadri of Windhelm is a Speech Trainer, and also owns a pawn shop.
• Lastly, the following trainers are normally not merchants, but can be married, which will cause them to open a small store.
  • Vilkas (Two-Handed)
  • Aela the Huntress (Archery)
  • Njada Stonearm (Block)
  • Farkas (Heavy Armor)
  • Dravynea the Stoneweaver (Alteration)
  • Scouts-Many-Marshes (Light Armor)

Dawnguard

Dawnguard adds three new trainers who are also merchants.

• Ronthil, a Speech trainer in Volkihar Keep, who sells fairly standard magic-related items
• Florentius Baenius, a Restoration trainer who can be recruited into the Dawnguard, and sells some assorted ingredients, potions, and a couple of spell tomes
• Sorine Jurard, an Archery trainer with the Dawnguard who mostly sells crossbows, bolts, and the like.

Dragonborn

Dragonborn adds three new trainers who are also merchants.

• Milore Ienth, an Alchemy trainer in Raven Rock who is also an apothecary
• Talvas Fathryon, a Conjuration Trainer in Tel Mithryn who also sells tomes and soul gems.
• Neloth, an Enchanting trainer in Tel Mithryn who also sells staves and soul gems.
  
  

Now, this all may be a very large list, but it's unfortunately not a very promising one. The best choices for selling goods to are generally Fences, as they have the most money (depending on your Thieves Guild side job completion), and also buy anything without need for the Merchant Perk. But the only Fence who is a Trainer is Ma'jhad, who is part of a Khajiit caravan; if you find yourself needing to wait around for his gold to replenish at any point, he will likely be in a different city when you're done.

Your next best bet is probably going to be a blacksmith, as several of them have a thousand gold (including all three who are also trainers), and they buy many of the items you're looting off enemies, even without the Merchant Perk.

Going through the list of trainers on The Elder Scrolls Wikia

Skyrim

• Lami - Alchemy adept trainer; alchemy merchant.
• Arcadia - Alchemy expert trainer; alchemy merchant.
• Babette - Alchemy master trainer; alchemy merchant ¹
• Tolfdir - Alteration master trainer; mage supplies (esp. alteration).²
• Aela the Huntress - Archery expert trainer; general goods (spouse) ^3,M,F
• Njada Stonearm - Block expert trainer; general goods (spouse) ^3,M,F
• Phinis Gestor - Conjuration expert trainer; mage supplies (esp. conjuration) ²
• Sybille Stentor - Destruction expert trainer; mage supplies.
• Faralda - Destruction master trainer; mage supplies. ²
• Gharol - Heavy armour expert trainer; blacksmith.
• Drevis Neloren - Illusion master trainer; mage supplies (esp. illusion). ²
• Scouts-Many-Marshes - Light armour adept trainer; general goods (spouse). ^M
• Grelka - Light armour expert trainer; blacksmith.
• Athis - One handed expert trainer; general goods (spouse). ^3,M,F
• Colette Marence - Restoration expert trainer; mage goods (esp. restoration). ²
• Ghorza gra-Bagol - Smithing adept trainer; blacksmith.
• Balimund - Smithing expert trainer; blacksmith.
• Eorlund Gray-Mane - Smithing master trainer; blacksmith.³
• Revyn Sadri - Speech adept trainer; general goods.
• Vilkas - Two handed master trainer; general goods (spouse)^3,M,F

Skyrim: Dawnguard DLC

• Sorine Jurard - Archery master trainer; general goods⁴.
• Florentius Baenius - Restoration master trainer; alchemy goods.⁴
• Ronthil Speech adept trainer; general goods. ⁵

Skyrim: Dragonborn DLC

• Talvas Fathryon - Conjuration master trainer; mage supplies.^F

^{1: Requires affiliation with Dark Brotherhood. 2: Requires affiliation with College of Winterhold. 3: Requires affiliation with The Companions. 4: Requires affiliation with the Dawnguard. 5: Requires affiliation with the Volkihar clan. M: Only sells if married to the Dragonborn. F: Potential follower (eligible for free training exploit). .}

Whilst not all trainers are merchants, most are near merchants. There are possibly a couple of errors in the above either due to me skim reading the articles or to them being not entirely correct - some NPCs were meant to be trainers but are bugged, for example.