The most obvious and well-developed relationship is with Julia, with whom he has a love affair that ends with their capture and torture by the Party. It is made very clear from the beginning of their affair that the whole thing is conceived of as an act of protest against the Party:
Their embrace had been a battle, the climax a victory. It was a blow struck against the Party. It was a political act.
Indeed, the thing that Winston most admires about Julia is her apparent disregard for the Party and its teachings, which she always describes as "rot."
Another relationship that Winston enters into is with Mr. Charrington, the apparently kindly old man who rents Winston a room above his store. Winston and Julia take to using the room for their meetings, but it eventually turns out that Mr. Charrington is acutally a member of the Thought Police, and he turns the couple in.
Finally, there is O'Brien, the Party official that Winston, for some reason, believes to be a secret dissident. He cultivates a relationship with Winston, even persuading him that he is possibly a member of the Brotherhood, before supervising his torture in the Ministry of Love.