Most questions should have 1 and only 1 correct answer (matching, fill in the blank, put in order, etc.).
Paragraph or few sentence responses or plotting can be uglier.
Try to give partial credit where you can. Always give points for correct steps even if the final answer's wrong. If they get the final answer but their steps or logic to get there is wrong, give them some points, but not too many.
Obviously if they screw up part (a) by a factor of 2, but carry that extra factor through parts (b) through (f) and get everything else right (while including the factor of 2), they should only lose points on part (a). Also, stress this fact to your students so they don't get frustrated if they can't do (a), but the rest are doable (maybe even tell them to make up an answer to use for later parts, or in the question say 'use 5km for the rest of this question if you don't get part (a)').
In longer answers, you should usually reward for correct information more than you punish for incorrect information. With that said, if they say something really wrong or even contradictory to the rest of their answer, they should be penalized a decent amount.
Hopefully on your quizzes and exams you stress to students that they must write clearly and explain their steps and logic clearly. If you can't read their writing or understand what's going on, you should usually assume it's wrong.
Be suspicious: If you see similar, very wrong answers, flag the tests and compare their answers to other questions. Hopefully you can look out for cheating while the quiz/exam is actually going on, but you won't be able to see everything.
Talk (probably through e-mail) to students in your section(s) who performed very poorly (grades of less than 40% or 50%). They may be too shy to ask for help even if they know they need it!