Out of every superhero in the DC Universe, there are hardly two that are more different then Superman and Batman. Superman takes on crises as they arise. He is so fast and powerful that if a bad thing happens, he will shoot over and stop it so fast that whatever hapless villain was making a stir ends up in jail faster than light. Batman prefers a crushing, preemptive strike. If a crisis will happen, you can guarantee that the Dark Knight will have known about it and already have a detailed, workable plan formulated to stop it. Crises don't happen in Batman's territory-for the Caped Crusader stops them before they start. Given that these approaches are totally different, it's odd that these two would work together.
But Superman and Batman actually have their own team-that is how often they collaborate to fight evil.
Much of the distance is in power discrepancy. Superman can move faster then the speed of light. He is literally impossible to damage in any physical way-only magic and Kryptonite affect him. He has nearly unlimited strength-he has moved planets. He never tires and he has an array of other weapons. His heat vision can melt steel, he can freeze nearly anything with ice breath, his super breath can make airplanes stop in midair, and he can fly. He is nearly unstoppable.
Batman has no powers-he is merely a man. But Batman is honed to the physical and mental peak. He is one of the three smartest men in the DCU, along with Mr. Terrific and Lex Luthor. He also is one of the greatest martial artists alive, and he has loads of technology and money, including tools like Batarangs. He is a master tactician. This is why they need to take different approaches.
Although both have a different method (and they are often at odds), they team up all the time, and together, are one of the greatest team-ups in history-Superman/Batman.
The "team" started when President Lex Luthor declared both heroes enemies of the state and put huge warrants out for their arrest. His reason-a Kryptonite asteroid is heading towards Earth, and Luthor blames Superman. He offers a 1,000,000,000 bounty for the two heroes-so every hunter in the country-plus many heroes and villains-are suddenly gunning for Superman and Batman.
Superman confronts Luthor in the Oval Office and nearly kills him, and Batman lets him-but Kal-El steps aside and spares Luthor. They meanwhile have to find a solution to the asteroid problem-and the solution is the new Toyman, who builds a spacecraft that can destroy the asteroid. This poses a problem for Luthor, so he takes drastic measures. He injects himself with Venom and synthetic Kryptonite, dons a battle suit from Apokolips and tries to kill Superman and Batman himself. He is defeated and exposed-and it costs Lex Luthor the presidency. This is documented in Superman/Batman # 1-6.
They also saw the introduction of someone important in Superman's life-another Kryptonian. The Kryptonite asteroid lands when Batman detects life on it, and Kara Zor-El-Superman's cousin-is inside the ship. She becomes the new Supergirl. Batman is (of course) suspicious-he thinks it was too coincidental for another Kryptonian to pop in now. It doesn't much matter, for Wonder Woman abducted the girl to Themiscyra for training. All of this, of course, attracts the attention of Evil Incarnate- Darkseid. He wrecks Themiscyra and attempts to steal Supergirl so she can lead his Female Furies. Batman and Superman stop the evil Lord of Apokolips, and Superman angrily throws him into the Source Wall. Supergirl is saved. This is documented in Superman/Batman # 8-13.
Later, Mr. Mxyzptlk battles the Joker, who has tricked Bat-Mite out of his powers, using other characters as their pawns (much like what he did to Mxyzptlk in Emperor Joker). Superman and Batman fight a team of superheroes from an alternate universe called the Maximums (a analogue of Marvel Comics' Avengers series, more specifically their incarnations from the Ultimate universe, the Ultimates). Keeping the bargain he made in issues 8-13, Superman frees Darkseid from the Source Wall, and, double-crossed by Darkseid, becomes trapped in the Source Wall instead. Recruited by Bizarro, multiple versions of Supergirl rescue Superman from the Source Wall before they are all transported to an arena.
There, they fight Darkseid, Lex Luthor, and evil versions of Superboy and Supergirl in a battle that involves versions of Supermen and Batmen from many different realities. Bat-Mite escapes, and the two imps tie up all the loose ends and return all the participants to their point of origin. This is cataloged in Superman/Batman #20-26.
Additional Members and Their Stories
While Superman and Batman are the main stars of the series, many of their friends, partners, and sidekicks have teamed-up with one another to defeat the forces of evil. While not an ongoing story line like the main stars; these co-stars usually are seen once or twice in the series.
Dick Grayson and Superman
These events happen after Final Crisis. Before Dick became Batman with Damian Wayne as his Robin, Clark was still struggling with Bruce dying. While he wasn't actually dead, however, lost in time, Superman and the other heroes didn't know yet. While other heroes missed Bruce greatly, Superman was the one who missed him the most. Clark and Bruce were the worlds finest heroes, they fought one another, however also destroyed a whole bunch of crime together. With Bruce gone, Superman felt guilty for not being the "man of steel (or shield)" for his best friend. When Dick Grayson was beginning his early days as Batman, Dick was testing one of his pre-suits before his modern suit. Superman got very angry at this point, because he missed his friend Bruce, so eventually with the help of Wonder Woman, Superman adjusted to Dick being Batman.
Supergirl and Damian Wayne
This was a short team-up that had Kara Kent (Supergirl) teaming up with Robin (Damian Wayne). When there was a massive death count in Suicide Slums, Supergirl went seeking for a Master Detective. Mainly Dick Grayson or Tim Drake. However, instead she found Damian Wayne in a situation against Killer Croc. When Damian told Supergirl that Dick Grayson was with the Justice League Of America and Tim was following his own adventure, Supergirl was going to just leave Damian with taking Killer Croc to jail. But unknown to her, Damian surprised her with talking about the killings in Suicide Slum. After grudgingly listening to Damian boast in himself, Supergirl knew that she needed to capture the killer, and allowed Damian to work with her. Later in the Metropolis city Morgue. Damian and Supergirl were studying the bodies of the victims that were killed in Suicide Slum. The victims eventually led to Scarecrow, and the team (Damian and Supergirl) beat him with ease and took him jail.
Damian and Conner
In the not to distant future, Damian Wayne is Batman with Conner Kent as Superman. Superman and Batman (Clark Kent and Bruce Wayne) both died protecting the universe from an omega level threat. To replace them, Conner Kent has chosen to become the newest Superman and Damian Wayne as Batman. For a mysterious reason (probably something in the past before the deaths of Superman and Batman) Damian and Conner hate one another.
However, once every year, they join one another at midnight to honor the two most important/influential people in their lives, Clark and Bruce. Damian and Conner are portrayed in their late 50's to early or mid 60's. Damian states that he is training Terry McGinnis to be the newest Batman, while Conner is about to retire his cape to someone else.