Do a lot of graduate programs provide assistantships to their students (e.g. teaching or research assistantships)? How common is it for graduate students to be paid while going to graduate school?
This depends on many factors, including the discipline, the degree, and the institution. Some disciplines have a lot more research dollars to support research assistants. For example, scientifically oriented programs tend to have more money to support students than liberal arts degrees; PhD programs tend to have more money than terminal Master's programs; and, of course, educational institutions exhibit a lot variability in terms of their overall resources.
To answer your first question:
Do a lot of graduate programs provide assistantships to their students (e.g. teaching or research assistantships)?
In my experience, programs do not provide assistantships -- professors do. When you are applying to a program, whether Master's or PhD, your application is considered in relation to work that the faculty of that department have on the table for the next year or several years. For RAships this is grant-funded projects; for TAships these are large-workload courses.
In my understanding, just about all graduate programs (except not so much in professional degrees like business, law, medicine, and perhaps a couple others - although there are definitely exceptions) provide R/TAships. This is so for simple reasons that there is research and teaching to be done, faculty don't have time to do it all by themselves, and graduate students are the cheapest workforce with post-Bachelor credentials for an institution of higher education.
So, yes, a lot of programs provide these assistantships. You should assume the program you are applying to does. This is easily confirmed by reviewing the websites of current grant-funded projects run by the faculty (research groups/labs/centers) and seeing if they have graduate students on their rosters.
To the second question:
How common is it for graduate students to be paid while going to graduate school?
This largely depends on the size (for TAships) and/or amount of research activity (for RAships) at the institution and the department in question. The typical heuristic applies: engineering and engineering-like hard sciences tend to be better funded, philosophy and theater tend to be barely scraping by, and the social sciences are somewhere in between. (Of course there are exceptions to every rule, and you might find a truly flourishing social science program here and there, just as you might find a relatively cash-light hard science program.)
Beyond these generalities, it depends on the fit between you (your qualifications and goals) and the on-going projects and needs of the faculty who will be advising you and whose research you are interested in. Most likely you will at times have to reconcile the trade-off between seeking an RAship on a project you are not really interested in, or proudly sticking with doing exactly what you want to be doing while paying out-of-pocket or debt financing your studies. This is the nature of the graduate game, and THAT is indeed common. Good Luck!
Generally, if it's a research degree (i.e., a PhD or a research-focused Masters designed to lead to a PhD), some form of assistantship or fellowship will be provided to cover tuition and a stipend. The details including the nature, requirements, stipend size, and other perks will vary by field, institution, and department and sometimes even by how much a department or professor wants a particular student. The degree to which this funding or these assistantships are guaranteed — and the periods during which they are guaranteed — can vary as well.
If the graduate degree is a non-research masters (e.g., an MBA) assistantships will almost never be offered or guaranteed although they are sometimes possible. Most students are expected to pay.