In the TOS episode Assignment: Earth, the Enterprise is sent back to 1968 to conduct some historical research. There's not much said about whether this is unusual, and judging by the lack of talking about this, it seems perhaps this may be regular. So, is there any evidence that shows us that time travel was commonplace for the Federation in the 23rd century, or was this an exceptional instance (and if so, why wasn't there mention of this being so special)?
On Memory Alpha, we learn that The Enterprise seemed to have a unique time-travel ability at the time the slingshot effect was used:
In the novel Forgotten History, it is revealed that, initially, the Enterprise was the only ship capable of performing a slingshot maneuver due to its engines being exposed to a unique set of conditions during their jump back in time after a cold start in "The Naked Time", resulting in the newly-formed Department of Temporal Investigations claiming the engines during the ship's refit to create a timeship.
It goes on, however, to say:
However, the knowledge of how to make any ship capable of performing a slingshot maneuver was revealed to Kirk by Agent Lucsly of the DTI when he was forced to work with Kirk during a malfunction of Timeship Two, which trapped them between 2273 and 2383.
Obviously something changes by the time of ST:IV, as they use the same slingshot effect (AFAIK) to time travel also.
So, in summary, it was not commonplace in the 23rd century - only The Enterprise could.