With respect to graduate school funding, many/most PhD programs are indeed funded, often – but not always – through TA-ing or working as a research assistants. (This type of funding may be more prevalent among public universities than it is among private unis because some privates will grad programs without making students work, but YMMV). Sometimes the length or amount funding will vary (my husband’s department will sometimes fund incoming grad students from anywhere ranging from one to four (or more) years, depending on how many students are being accepted and how strong the candidates are.
One thing to note: as a grad student at a public uni, the rules for establishing in-state residency and being charged in-state tuition rates at public unis differ from those of undergrads. Generally, after the first year, assuming that you can prove full-time residency through various means such as having rented an apartment, having gotten a driver’s license from that state, can prove employment, registering to vote and putting utilities in your name, you will be considered to be instate for tuition purposes.
Terminal Masters programs are generally not funded, with some exceptions (e.g., through fellowships and/or TAships for an MFA program when the masters is the highest level that can be obtained in the field).
There may also be fellowship funding or merit aid for some graduate programs.
As to the logistics of applying, most students don’t visit before applying and generally aren’t given “tours” nor are they “marketed to” in the same way as undergraduates are. Applications go to the individual departments and are reviewed by faculty committees – not admissions staff. They want a coherent statement of purpose – why you want to study what you want to study, what your objectives are, and what you can bring into the program; unlike undergraduate admissions, they don’t want to hear an interesting anecdote about shopping at Costco, how you overcame adversity, or a letter to your roommate. Recs from professors and test scores often matter more than GPA, but again, that depends on the program (professional schools like law and medicine may focus more on GPA than more academic fields, but again, it depends).
Once accepted into a program, you can (and should) visit campus, sit in on classes, and meet faculty and students in the program. Often the departments will arrange for you to be housed with a grad student in the program. Don’t expect to be courted at organized events (e.g., “admit week”) the way that undergraduates are. You will likely be working directly with department staff in setting visits up.
As for housing, it depends on where you will be studying. In more affordable areas, grad students – especially older ones – will prefer to live off campus. In high rent areas (think Palo Alto, New York City, etc.) on-campus housing might be more desirable.
As for working outside the program, that would obviously depend on the circumstances, the funding, the time commitment for being in the program, etc.
As to whether most student go directly into grad school, that depends entirely on the program. Some programs prefer people that have real-world experience (e.g., MBAs, Engineering), others will prefer students still in the academic trenches.