I'm thinking of starting my own campaign with a few of my friends with me as DM, primarily because I wasn't extremely impressed with the campaign my friend was DMing. When I started thinking about the things I really didn't like, one of the biggest ones ended up being combat. It's possible that our DM just wasn't creating very interesting encounters, but it seemed like combat always took way too long, was just a bunch of us sitting around rolling dice, and every single one was the same.
How can I keep a campaign interesting and engaging with minimal combat encounters? Instead, the game could be based around Diplomacy, exploration, puzzle solving, gathering intelligence, stealth, etc, but I'm having trouble thinking of non-combat quests and fleshing them out to the point where they are actually playable. Some combat is okay, but I don't want the quest to revolve around it.
Has anyone here run games like this? What do you usually base the quests around? How can I keep things interesting and still allow the players to advance their skills? Can you outline or link to any sample quests that have a minimal amount of combat encounters?
NOTE: Although fixing the issues I stated with combat could help this situation, that's not what I'm looking for here. I will probably ask a different question about making combat encounters more bearable so this one can be focused on running campaigns without combat.