When you grade an assignment, don't expect to be perfect. In fact, don't even try to be perfect. No matter how much time you spend grading you will make mistakes. Even if you don't make mistakes, students will come to you with complaints about their grades. Strive for a consistent and reasonable policy - consistency in grading each problem, and consistency in grading each assignment. Tell the students what your criteria are and how you assign partial credit. Students are less likely to complain if they understand what is expected of them.
Grade one problem at a time across all assignments. This helps you be consistent and is actually faster than grading one assignment at a time. Write down your criteria for grading each problem and try not to change it as you go along. Read through a few answers first before establishing the criteria. Make notes about the most common mistakes and point them out in your published solutions or in quiz sections.
Give partial credit, but don't use too fine a scale. Not only does it lead it inconsistency, but it also promotes point-grubbing (students coming in to beg for more points). A ten-point scale for an assignment is typical. You probably don't want to assign more than four different point grades to a single problem. Keep in mind the minimum correct solution and criteria for deductions of points.
Grading programs is quite different from grading written assignments, quizzes, and tests. Code is typically tedious, complicated, and voluminous. Create input files that test the functionality of the program and allow the students to test their own programs with the input files before turning them in. Grading criteria can include successful compilation, correct output on inputs, functionality, performance, implementation, style, and documentation. Ask the instructor what he/she believes is important.