I play either the career or the sandbox, usually alternating between. I use sandbox to test out designs, learn how parts work, check out the new aerodynamic features, and just plain play around. For example, with the new aero-heating and parachute deployment limitation “features” in KSP 1.0, I had a hard time safely retrieving vehicles. I spend a lot of time in sandbox testing out both vehicle designs and re-entry procedures. I’m not sure I’ve found an optimal solution, but at least I’m able to come home safely most of the time!
Career offers a bit more challenge in that development is constrained in a somewhat logical sense (funding and technology limits), but I don’t really like the contracts or the science system. Some of the contracts don’t make sense and are incredibly difficult (activate a jet engine at an altitude between 33,741 and 33,821 m while in a suborbital trajectory at a velocity of between 2,319 m/s and 2,407 m/s = wut). The “game” of driving around the Kerbal Space Center and racking up a couple hundred science points by doing experiments around each of the buildings also seems forced.
So I typically play career until I get annoyed and go back to sandbox.
As for mods, I prefer the vanilla configuration, although I do use Kerbal Space Engineer to get vehicle delta velocity capabilities by stage - it’s certainly a heck of a lot easier than calculating it by hand!
For activities, I like planning complex maneuvers and executing them. There’s also something special about bringing a Kerbal to a foreign planet, doing some romping, and then bringing them home safely again. I’ve been to Mun, Minmus, and Duna and back, safely. I’ve had Kerbals in orbit around Eve, and brought them home, but still haven’t mastered the descent/ascent/return part of that mission.