greenfrogs

It seems you come from abroad? 
I remember being in the same boat as you, trying to figure out what's what. 
Back in the day, I did what @TMP suggested: identifying professors and programs simultaneously. Many professors disclosed early on that their programs did not offer "good" funding for international students, so they were are no. I also crossed out programs that required teaching in my first year because I didn't want to deal with grad school + new country + second language + teaching. 
In the end, my decision came down to who offered the best net package (not only tuition waiver but also stipend, health insurance, possibility of research funds, etc). 
I must also add, there are a lot of hidden fees for international students. Remember you might need to renew your visa at some point and this costs money and travel. In addition, while health insurance might be good (ours is pretty decent), you will still need to pay out of pocket. So take this into account when comparing funding packages. 
Best of luck!

I want to add that while many schools don't offer good (or any) funding packages for master's students, there are some that do.
I'm doing my master's now, and it's fully funded with a livable stipend and some funding for research. So if you're not sure about doing a PhD but may want to do a master's first, you could look around and find schools that would offer you funding. I had several offers from master's programs all of which were fully funded. Don't look for these offers at top-20 schools, because they generally only offer funding to PhD students, but check out schools that are lower ranked but support their MA students. 

I agree.  First step, identify who you'd like to work with and look at different programs.  Then determine what programs do offer funding (preferably 5-year packages). Eliminate the rest.  Entering in my final year of the PhD, I can tell you it's not worth paying a dollar toward what should be a basic funding package (tuition waiver, some fees, living stipend, subsidized health insurance).  There are enough hidden costs as they are and they add up (i.e. your university may give you only $500 for conference travel but flights, public transportation/rideshare, hotel, food, conference fees, etc. might add up to $600...).

Different programs do different things, but I will say this: academics can only be rigorous if they're funded, and a dumb student with lots of financial support will almost certainly write a better dissertation than a smart student with no support. If a program does not offer a livable stipend, cover tuition, provide affordable health care, and provide access to pots of money for travel and research, it is not worth applying to, never mind attending. 

My process was to identify the professors with whom I wished to work, and then to narrow down the list by excluding programs that offered insufficient resources. Aim for a final list of 4-6.