I'm Canadian and after having done two graduate degrees in the US and having consulted with my own immigration lawyer here (my attorney is in Washington, DC), I would suggest that you do the same: that is, get yourself an immigration lawyer here in the United States once you have been admitted to a law school.
I'm not exactly sure how it works with law schools, but if they abide by regular US university rules (via the Department of Homeland Security) , then you will have to enter the US on either an F-1 or J-1 Visa. I would imagine that unless the law school is giving you a complete scholarship - and when I say complete, I'm talking paying for your food, housing, tuition, books, everything - or a major agency in Canada is footing the entire bill - then you would likely enter on an F-1. For the F-1 you must be able to prove that you have enough money to support yourself in the US throughout the duration of your studies. I had to provide banking documents and documents from my RRSP and other investments in my applications.
After you are done your studies you have what is called a one year "Optional Practical Training" visa that you can apply for providing that you can prove you have a job offer for a year. After that, you need that immigration lawyer to lead you in the right direction. He or she will talk to you about what to do after you get a job offer from a US law firm re: H1-B visa or a TN-visa. Beware - the TN visa through NAFTA is only for those who do not plan to permanently reside in the US. Presumably if you are doing law school you are planning on living here permanently. In that case your lawyer will probably advise the H1-B and then later suggest an application for a green card. It is a lengthy and expensive process, but not an impossible one. Do your homework and you'll see what I'm talking about. Immigration lawyers aren't cheap so be sure you prepare yourself for that cost. Again, do your homework and you will see. Hope this gets you started in the right direction.