This is one of those topics that I'll have to complete in stages, as the topic is so vast. According to the Grand Comics Database, Professor Carter Nichols, who came up with the concept of time travel in the Batman canon, appeared in 31 stories prior to 1964 (when he disappeared like much of the Batman family).
When he first appeared it was apparent that his "method" of sending someone through time involved nothing more than hypnosis. From Batman #24's It Happened in Rome:
And a few panels later, alarmed that Bruce has not returned, Dick asks to be sent back to rescue him. We see the technique for the first time:
Now the concept of people being hypnotized into the past is a bit silly, but let's remember that John Carter of Edgar Rice Burroughs' Mars novels used to travel to Mars by doing little more than wishing himself across space. However, it certainly appears that the editors, writers and artists recognized this weakness, and while the hypnotism was never quite discarded, they did surround Professor Nichols with scientific-looking equipment that we could imagine was really doing the heavy lifting.
That initial story featured Ancient Rome and included a sequence with Batman in a chariot race (probably inspired by Ben-Hur; the book came out in 1880 and was one of the best-selling books of all time). The next appearance was in Batman #27's All for One, One for All and clearly was an homage to The Three Musketeers by Dumas. The Dynamic Duo went back to King Arthur's Court in Batman #36.
Heh, looks like he's got time-travel rays shooting out of his eyes there.
One interesting aspect of the early time travel stories are all the "doubles" for characters from Batman's then present-day adventures. For example, in It Happened in Rome, Batman meets the Jester, who strongly resembles the Joker, although in a twist the Jester befriends Batman and saves his life on one occasion. While in Batman's King Arthur story:
In fact, that tale continues the long-running motif of Batman "accidentally" letting the Catwoman get away.
Another common theme had Bruce and Dick traveling to the past to solve some mystery clouding the reputation of someone in the present. For example, in Batman #52's Batman and the Vikings, Bruce is startled to find a stone carving of a man bearing a distinct resemblance to him. Even more surprising is the inscription:
Although the man in the carving is no ancestor of Bruce's, he feels compelled to learn the full story. In the end, he finds that the part of the inscription that was cut off tells a different story:
This theme was used over and over again in the Carter Nichols stories. In Batman #93, an archaeologist finds his reputation ruined when he locates a stone carving showing a T-Rex chasing some cavemen. Since dinosaurs died out well before the first caveman it appears that he's been taken in by a fraud. But Bruce and Dick go back in time and discover that the T-Rex had been frozen in some ice, and only came to life briefly when a fire melted the block. They found the rest of the carving which showed the dino in the cube. I talked about that story over at Silver Age Comics.
In Batman #89 the Dynamic Duo travel back in time and clear Commissioner Gordon's great-grandfather:
And in Batman #99 Batman is confronted by a reporter with an old newspaper that alleges he used guns one time in the old West, but it turns out to be a gent with a similar name:
In a twist, Batman and Robin brought sci-fi writer Jules Verne forward into the present in the Return of Mr Future from Batman #98. A crook has stolen Verne's greatest invention, a sonic gun. Verne has thought of a way to defeat the weapon and must help Batman design and create it, defeating the criminals. Afterwards, they show him some of his predictions that came true:
Cute story. One of my favorite Batman stories of all time is the Second and Batman and Robin Team from Detective #220. In that story, medieval scientist Roger Bacon discovers Professor Nichols secret and sends two young men who look remarkably like Bruce and Dick forward in time:
In Batman #67, Robin travels forward in time to help the Batman of 3051, whose junior partner has broken a leg:
Although this is not a Carter Nichols story, that panorama is so beautiful that I couldn't resist including it in this post.
Much more Batman time travel to come!
Update: Also, check out Bill Jourdain's column on Professor Nichols from last year.