Without going through every episode and tallying them up, a few trends can be confidently stated:
- Riker leads - Our very first introduction to Riker as a character, in the pilot episode of TNG, has him putting his foot down about Picard not going on away missions. In many ways, Riker filled the narrative space that Kirk had held in TOS, much more than Picard did: Riker was the "adventurer" archetype, the handsome young heartbreaker who loved a challenge. Riker is most often in charge of away missions throughout TNG.
After that, TNG takes a much more specialized approach to away missions (and really most ship operations) than TOS did. Away teams are less traditional, and more tailored toward the needs of the operation:
Data does general science, but Geordi does tech - Most away missions will have one or the other of these two, depending on the nature of the mission. If it has to do with repairs or analysis of equipment or a ship, Geordi will go, otherwise Data tends to handle it.
If there's danger, bring your Security Chief - First Yar, then Worf would traditionally join away missions into dangerous or hostile territory. If it's a really scary spot, they'll bring some muscle with them.
When in Rome, bring Troi - Diplomatic missions almost always had Troi among the team, so they could use both her Betazoid abilities and her general tact.
Doctors only when needed - Pulaski and Crusher are occasionally brought along "just in case," but generally they are called in only when someone (Enterprise crew or local) has already been injured.
Picard for ceremonies, pleasure, or archeology - The Captain (per Riker's insistence) tends to stay on the ship, but he can reliably be seen on-world if there are ceremonies or events to attend, if he's on vacation, or of there's an awesome old-thing that he can't resist checking out. That ol' amateur enthusiast, Picard...
This is all a part of TNG's generally more "professional" style: TOS had more of a free-wheeling adventuring structure, while TNG played much more into the naval traditions and other daily necessities of serving on board a starship. As such, the away missions were made up less of "our heroes," and more of the officers needed for a given mission.
Also, Roddenberry and the showrunners took a number of steps to minimize the "redshirt effect," where every away mission would involve a few untouchable stars and some cannon-fodder. For example, they killed Yar because the actress wanted off the show, but I've heard accounts that they did it on an away mission to keep the audience from thinking that anyone was safe during such scenes. Another part of that effort was shaking up the lineup of any given away mission, and sending more main characters on away missions to minimize the disposable extras.
But if you're looking for a specific count of each character's participation in away missions, I think you might need to spend some time on Netflix.