The Avengers were never terribly comfortable with the tendency of the Justice League to take teenage "apprentices" with them into battle. This stemmed from, ironically, Captain America and his loss of Bucky. Bucky was only sixteen when he started doing Special Forces work with Captain America and while he was almost twenty when he "died", the event left him thinking soldiering was better left to the adult.
Captain America's one conversation with Batman on the subject was doubly painful as Dick Grayson reminded him of his young sidekick in countless ways. Batman, in his usual polite manner, told Captain America where to stick it. Needless to say, the Teen Titans never quite felt welcome in New York as a result.
The Teen Titans would find this attitude completely subverted by the original X-men. Stumbling upon Charles Xavier's School for Gifted Youngsters quite by accident, there was the usual fight followed by the realization they were falling into the habits of the adults. Jean Grey and Donna Troy would become close, Dick Grayson would share leadership tips with Cyclops, and Wally West adored Iceman's jokes. Even Beast found a curious friendship with Aqualad as the mystic built a rivalry with the science-minded Beast.The exception was Roy Harper who came to Hawkeye at one point, mostly to spite his old man. Clint Barton and he became extremely good friends, the working-class hero having no small amount in common with the young man.
Clint was also surprisingly non-judgemental about Roy's drug problem and the two parted as friends. It would later be a great irritation to Green Arrow, Roy considered Hawkeye a surrogate uncle.
The Teen Titans would have several sparing matches with the X-men and it was a generally innocent time for both of them. Charles Xavier considered the Titans to be a perfect example of how mutants and humans could build a rapport. Spiderman, perpetually the butt-monkey of his early career, was eventually invited to join in the games as well along with newly discovered mutant Firestar.
The original Titans would break up soon after while the X-men would go on to be a more cosmopolitan school. When the New Teen Titans were assembled by Raven, Kitty Pryde attempted to join them only to find them less than excited about a thirteen year-old girl hanging around them. Amusingly, as time passed, Kitty Pryde and the New Mutants would end up visiting on a regular basis.
The greatest benefit from this encounter was for Ilyanna Rasputin a.k.a Magick, who had spent her entire developing life being horrifically abused by the demon Belasco. Raven, sensing the insanity the girl was hiding, helped her confront her demons.
Taking up Azarath meditation, Ilyanna managed to dramatically scale back the corruption she'd suffered. Raven would be stunned when, during the hour of her corruption by Trigon, Ilyanna would teleport out of nowhere to stab her with the Soul Sword and help her regain control. As such, Raven's death and sacrifice were never required by events.
The Titans would repay the New Mutants during the Inferno event, helping battle back the hordes of demons until they reached the heart of the darkness. Not only would Ilyanna be prevented from reverting back to childhood but Raven actually managed to cleanse Jean Grey clone, Madelyne Pryor as well. Traumatized by the revelations she'd experienced as well as torture at the hands of Genoshans, Madelyne would leave the lives of the X-men and briefly served as a member of Doom Patrol.Raven and Ilyanna have a co-dependent friendship because of these events. When Ilyanna is feeling like Limbo is about to overwhelm her, she invites Raven to help slaughter demons for a few relative decades. This has allowed Raven to work out some of her anger issues and keep the darkness relatively under control.
In Hell, they're known as the Devil's Daughters and Limbo is off-limits to all would-be conquerors less than Dormammu or Sataanus.
Ultimately, the friendship of the Titans and New Mutants was severely tested by the arrival of the mercenary Cable. The brutality and murder Cable advocated after spending his adult-life in Apocalypse's future convinced the Titans there was nothing good to be had by listening to him. Unfortunately, team-leader Cannonball chose to follow him anyway and any association between the two teams collapsed.
Wolfsbane and Beast Boy still exchanged letters, however.