PhD in the sciences, particularly biosciences, chemistry, physics, and engineering: Any top 50 program will pay your tuition plus a stipend for living expenses. The stipend ranges from 20-30K depending on field (lab sciences, especially biosciences and engineering tends to be higher, more on the 25-30K range) and cost of living in the area. My program paid for health insurance as well. You are typically treated as and employee or as a student depending on which is more convenient for the academic institution. There are also fellowships (such as the National Science Foundation Graduate Research Fellowship) which are considered very prestigious. Typically, if not on a fellowship you are either a research assistant or a teaching assistant to cover your stipend.
PhD in the more technical social sciences: Psychology and economics tend to be structured like the sciences. Economics stipends are on par with science stipends (probably more the 25-30K range), psychology may or may not be lower depending on the program. Psychology that are tied to neuroscience programs are structured more like science programs. Psychology programs are extremely competitive. Anyway, typically your tuition and living expenses will be covered. Again 20-30K, depending on cost of living of the area.
PhD in Humanities (English, Philosophy, Linguistics), History, Sociology: Depends. Fellowships cover some of these, but often tuition and stipend are covered by teaching assistantships. If you can't get your tuition and living expenses covered, you really should not do these PhDs because the job market for academic philosophers is so rough. Most people I know in well ranked programs are funded. Typically stipends are a bit lower, think 18-25K.
PhD in Education, School Psychology, etc. Typically these seem to be funded through fellowships and teaching assistantships to me, similar in range to humanities PhD programs.
Keep in mind that after the first couple years of a PhD, it's not really school. It's more like a low paying job/apprenticeship, which is in part why the pay structure is so different. It is also a long haul; 5-8 years.
Masters degrees:
MAs/M.S. In the fields listed above. It depends; sometimes you will have to pay, sometimes you get tuition + a stipend for a terminal masters. I have a friend who recently got a masters in philosophy and worked as a teaching assistant. She got tuition waived + 12K for living expenses. Her parents paid her rent, otherwise she would have had to take out loans to cover the rest of her living expenses (no dorms or anything, so she had to pay market rate for rent). Fellowships to get these can be competitive. I believe engineering occasionally will be funded as well; my cousin had tuition + a stipend covered for structural engineering. Typically these are only 2 years. Another way to get a masters without debt is to quit a PhD program where you are funded, known derogatorily as "mastering out." I have a friend who "mastered out" of a computer science PhD program and now has a very lucrative career as a software engineer, so in some fields it is not looked down on as much as in others.
Professional Degrees: Not so easy to get financial aid/tuition covered. Typically if you are over 22, you can declare yourself as an independent (you should look into the details about how this impacts one's ability to stay on parents health insurance and such, though. Often student plans are reasonably cheap), so no, parents income is not considered.
Law School: There are a handful of scholarships, mostly at second tier schools. It's risky to go to a second tier school, though since the job market has gotten so tight for law. Most of the people I know who go to law school take out loans (up to 200K+ of debt) or get help from their parents. Some cover some living expenses/some of their tuition through working for a few years before going to law school. Afterwards, they scramble for a Big Law associate job to make loan payments or they enroll in federal income-based loan repayment programs at a lower paying, less prestigious job. If you are in state for an excellent public school (i.e. Michigan, UVA) that can be an option to lower the cost through in-state tuition. Another option is that sometimes if you get a PhD in a STEM field and then work for a company, they will pay for your law school so you can become a patent litagtor. This happens very rarely, but I have heard of it happening on occasion.
Medical School: Expensive, but one of the few routes these days where you are guaranteed at least a decent salary/career stability when you finish. There are some scholarships, but not very many. The situation is similar to law school; take out debt, work for a couple years to pay for some of your expenses so you can take out less debt, or have your parents help you out. Not unheard of to take out 200K+ of debt. The idea is your earning potential increase will make this investment worth it. One option for medical school is to go in state where the tuition is lower, and this is less of an impediment to your career than law school, since doctors are in demand (maybe for the most competitive, highest earning specialties, it matters if you go to a top school, but if you want a decent paying career you don't need to go to Harvard). Another area where your medical school tuition is paid for is if you go into an MD/PhD program geared towards training you to be a clinical researcher. These programs can be 8-12 years long and people end up being very bitter when they finish them since after that slog they still need to do residency. They are also extremely competitive.
MBA: Typically you need to pay for these. There may be some scholarships, but I haven't heard of anyone getting one. Typically it is a combination of loans, savings, and parental help. For the MBA, since most prestigious programs prefer several years of work experience, you can save to help with the costs. I have one friend who worked as an actuary for about three years and then went on to Stanford business school. She had a crazy entry level salary, but lived a lifestyle of someone with a much lower salary because she knew she would go to business school in a few years. Occasionally, if you work for a while, your company will pay for it. Not worth it unless you can get into a top school (Sloan, Wharton, Stanford, etc.) IMHO.
MPh: Some scholarships and fellowships exist for these, but not as likely to have a stipend. Mostly you have to pay full freight through loans.
Teaching MA/certification. From what I hear these typically end up being about 10K out of pocket. There are a lot of programs to expedite the process by working in a title I school, etc. so it ends up being 1 year instead of 2. There are also loan forgiveness programs in public schools.
That's all I can think of. I am still in my 20s, so I think I have some perspective on the grad school/professional school scene having recently been through it in my field, and having watched friends in a variety of career paths go through it.