Zoro lifts weights, Sanji cooks, Robin reads, Ussop spends time tinkering and telling stories, Chopper makes and studies medicine, Navi manages finances, accounting, and inventory while navigating and drawing her maps, Franky maintains the ships engines and machinery, Brooke plays music, and Luffy must do the all-important task of goofing off and impulsive decision making.
Looks like the Strawhats don't have time to "train" their skills, but they use what they know and learn from everyday life and their adventures to ingeniously develop and create moves.
Luffy got his strength, reflexes, and instinct from Garp's training when he was young. As he goes through hardship and challeges, his determination and fortitude makes him stronger in strength, reflex, instinct, intuition and spirit.
Zoro gained his strength from endless training and determination since childhood. His instincts and intuition gather from experience. His ability to cut through the impossible is Haki, and not realistic.
Nami had no martial training ever. Her abilities come exclusively from her experience as a thief, her gift of weather intuition, and her knowledge of the physics of weather.
Ussop has his intellect and his prodigious dexterity. His inhuman stamina comes from his father’s heritage.
Sanji learned his cooking and fighting skills from Zeff, probably simultaneously, and fusing the principles of his kicking art with cooking. Zeff probably also trained his physical strength. His increase in physical strength as he adventures is likely from his mental strength and resolve.
Chopper has his science and his intellect. He has high muscular strength in heavy point because of his build.
Robin is smart. Her experience as a fugitive from her childhood taught her to be wary and observant. However, her capabilities are mostly from her intellectual capacity.
Franky is intelligent as well, despite being freaky. Using his knowledge of mechanics and machinery, he turned his body into a cyborg. How his body survived the process is unknown, but doubtlessly unrealistic, and nonetheless insignificant. All that matters is that he’s a war-machine.
Brooke was formerly a royal guard, and during those days, he developed his swordsmanship. However, he has a penchant for music, so he developed his musical talents as well. That somehow translates to him being able to attack using music. However, in reality, you can’t play a violin well using a sword, no matter how much you develop your technique.