I think that the best approach might depend upon what type of graduate degree you want to get.
As one example, PhD’s are usually fully funded. The situations that I am familiar with the student worked (or is currently working) part time either as a research assistant (help professors with research) or as a teaching assistant (help to teach an undergraduate course, possibly by grading homework or answering student’s questions), and in return had the school pay their tuition and fees, and in some cases health insurance, and also give the student money that was enough to live on. In all of the cases that I am familiar with the university provided the financial offer at the same time that the student got the offer of admissions to the program. I know of one case where the offer of admissions did not actually come in until early April (more often it would be slightly earlier), so a student applying to programs right now might possibly have a while to wait.
I know some people who got a master’s degree before applying for PhD programs, and some other people who got accepted to very good PhD programs with just a bachelor’s degree plus relevant experience. In the cases that I am aware of where a student was accepted to a PhD program with just a bachelor’s degree, their application was quite strong in other ways, such as having high undergraduate grades and good research or other relevant experience, along with strong references, and very good essays related to what the student wanted to study and why (and how this fit with the research already being done by some of the professor’s at each university they applied to)
I have heard of cases of students getting a partial scholarship to offset part of the cost of a master’s degree. Often however the student just pays the full cost of the master’s degree. Doing this part time while also working can take quite a while, although I have heard of this happening. In Canada, for students who are Canadian citizens, these are often also fully funded by an assistantship. In some fields at universities in the US there are some one-year master’s degrees, which limits the cost simply by having the whole thing not take very long.
Personally I went with the “one year master’s, do it quick, and get back to work before you run out of money” approach. It was quite tight on the “run out of money” front, but I did pull this off with a loan from my parents.
MD’s and DVM’s at least from what I have heard usually take so much concentration and so much effort that you just have to do it full time. 40 hours per week is not even close to enough time to commit to the effort to get either of these degrees.
I do know someone who is getting a DVM and had a part time job for the first three years of this in a drop in clinic. However, the $$ amount that they earned in their part time job was very low and really not the point. The point of the part time job was to get clinical experience that augmented their education. As an example, they did see some questions on exams related to diseases or injuries that they had already seen in actual patients in the drop in clinic. The amount they were paid for this job really did not amount to enough to matter compared to the cost of the program.
In all of this, the one case where it might make sense to work nearly full time and also study might be a master’s degree. However, it could take multiple years to get a master’s degree this way. I would be inclined to ask each university about whether they would support this. At least from the university’s point of view, there would seem to be a huge difference between having a full time student, versus having a student that is taking one class at a time for many years while also working.
And “The bank of Mom and Dad” has been a factor for a lot of the graduate students who I have known, at least for any program other than a PhD.
This might be the most important question.