Simply put, critical thinking skills involve establishing the credibility of information sources, challenging the assumptions behind conclusions and reasoning, and confirming the consistency of your own logic and reasoning. It is the scientific method applied to thinking itself. Therefore I will offer some guidelines which are generally applicable, rather than specific assignments limited to particular subjects.
Typically, assignments which build critical thinking skills are of the 'open-ended question' variety; they often are open to wide interpretation, and have no deterministic 'right' answer. For instance, If I ask 100 people what 6 times 7 is, they will all give 42 as an answer (with the possible exception of the math nerd who asks, "What number base?"). If I ask the same 100 people why Thomas Jefferson made the Louisiana Purchase, even though it was contrary to his current political philosophy and stated policies - then critical thinking skills become engaged.
The respondent (student) must first consider the facts as known. He must research to add to his knowledge of the facts, and to seek validation of them. He must consider whether the sources of this information were biased or not. He must challenge the basis of my question (What was Jefferson's political philosophy? Was this action inconsistent with that?). To answer the question, he must make an argument for or against, supported by facts, or must challenge the question itself. And in this process, he must continually consider, open-mindedly, whether his own reasoning is valid or not, being willing to alter his conclusions and rethink the question when he finds it to be invalid. Almost no-one thinks this way naturally: they must be taught this method, and taught how to apply it. And very few students (in American public schools, anyway) are taught so. If anything, critical thinking skills are taught as individual, isolable skills (like how to research), sometimes incorporated into the regular coursework for other subjects by the instructor, but rarely as a separate, integrated subject that is generally applicable to all courses of study.
Therefore, the student himself, or the student's parent, can learn about, and learn to apply, critical thinking methodology to the student's regular homework on their own initiative. This will usually be a learning experience for the parent as well. At first it will seem like a lot of extra work - and it will be - but it is a fundamental tool that returns incalculable dividends in a lifetime of clear thinking and learning abilities for the student. It will prepare the student to much more easily write essay answers and research papers; it will train the student's mind to form and defend his argument much more effectively (a valuable life skill); and it will give him an independence few students possess. That is, he will become a confident, independent learner, able to confirm his own studies, not merely one of the 'sheep-like' consumers of education, receiving only what the instructor and the school dish up. He will have the tools to confidently make his own way, chart his own course.
Is it hard to learn? Not exactly. It is relatively simple and logical, but it requires time and diligence and practice. Following this course, it will become much easier - second nature, if you will - to apply these concepts and techniques naturally and automatically.
So how can one begin to learn and utilize this valuable technique? First, learn about critical thinking skills; for beginners, or younger students, the most basic precepts can be incorporated first, building the skill set over time as more proficiency is achieved. Second, look for ways to apply these techniques to the student's regular homework: go beyond the 'facts regurgitation' normally asked for, and create essay-type questions, then defend the answers. If you are very motivated, you can also incorporate these techniques by discussing news stories and current events.
Critical thinking is solid work, especially at first. But the rewards are enormous, and the practice of these skills gets easier the more it is done, eventually becoming incorporated into the student's habitual way of thinking. First principles become second nature.