There's several types of away teams in Star Trek. Most of them don't involve the captain, and would be quite boring to follow onscreen.
Starfleet regulations recommend against the Captain going on away missions. The first officer is the one who's supposed to be doing that. Picard and Riker have both been reminded by Data and others about this regulation, but at least one of the two have said that the final decision was the captain's anyway - and they want to go join the away team. Stretch their legs, get a firsthand view, and so on.
Starfleet captains are also generally trained in diplomacy and first contact, so having the captain nearby on that type of away mission could be considered a good thing.
Later on in Voyager, Janeway didn't go on nearly as many non-diplomatic away missions as Kirk, and once even commented to one of her officers about how much freer captains were in Kirk's day. She sometimes wishes she lived in that era of "cowboy diplomacy".
EDIT because I missed out on the non-Captain part of the question:
I think the answer lies in two places:
- All the different types of away teams
- In Star Trek, unlike real life, the heads of the departments aren't managers - they fully understand their field, and are the best of the crew in that field that are on the ship.
So when something goes south, you generally would want your best people on the problem. Geordi went on away missions where there might be structural damage because he was the best at understanding the possible interaction with the ship's systems, the various head doctors when there was injured because they had the most range of experience with alien species, Spock's encyclopedic brain held helpful information on just about everything, and so on.
Due to the different types of away teams, it's very likely that between episodes the heads of the departments don't go on many away missions: There's no real danger/urgency, so no need for the best-of-the-best to be there. Likewise, there's no real reason for an episode centered around those events.