The first time I played Skyrim was with a female Nord, and I was able to train One Handed and Smithing with Amrem and Ghorza. The second time I played with a female Imperial, I beat their quests but they didn't offer me the option to train. After that I played with a female Redguard and another female Nord character and I was able to train with them. Does anyone knows why his happened?
This is all because patches. If you remember, in first versions of Skyrim you could take Faendal as your companion, train Archery with him and take your money back from his inventory. Later on this was removed. Same stuff with it. Also, some NPC's are eligible to train you only if you play certain species and gender. Read the NPC's description on wiki, you'll find the full info about any NPC in Skyrim.