Did this recently. The TN Visa (or permit) is by far the simplest way, but ONLY if you met the criteria. The process comes out of the NAFTA Agreement and hence applies only to Canadians and Mexicans; the very simple TN Permit process I used is available only to Canadians, I understand. It contains a specified list of professions, mostly university-degree-based. Basically IF you have a job offer that REQUIRES the professional qualifications (must be clear in the job description), and IF you can prove you have the qualification, you will get a TN Permit. For example, I am a Chemist by profession with university degrees in that field, and my job required a Chemist or Chemical Engineer.
Check out the INS websites for info on this. But the best source I found was my nearest US Customs Port of Entry. The people who actually do this stuff told me everything I needed to know. They pointed out the short easy way, for example. Although the INS website talks about applying for an TN Visa through their offices, it turns out the border guys themselves actually issue a TN PERMIT on the spot when you present yourself at the border, and this is exactly the same as a TN Visa. I talked to them ahead of time to be sure I knew what documents they would need. Then, at their suggestion, I FAXed all the docs several days beforehand for the official to check out, then phoned later to be sure it was all in order. When I arrived at the border all the papers were ready to go - ZAP!
A TN Permit (or Visa) is good only for one year. You must apply before that deadline for a renewal, with almost the same supporting documents (updated), including a letter from your employer showing that you are still employed in a job requiring your qualifications. But these permits can be renewed without limit, so you just keep on renewing as long as you work in the USA. If you lose or quit your job, you must leave the USA and surrender your TN Permit, since it requires that you have a qualifying job.
For other ways to work and live in the USA legally I can't say much. I know almost all have quotas per year, per country of origin, per job type, etc. There are no such quotas on the TN Permits. Also there are no requirements that a prospective employer demonstrate they were unable to find qualified Americans for the job, whereas other processes may have that requirement.
To open bank accounts, etc, you MUST get a Social Security Number. You MUST have it to be employed and pay taxes. And lots of businesses (e.g., electrical utilities and phone companies) use it in their account records. You get a special class of SSN because you are not a US citizen. So arrange to get that number as quickly as you can, so you can get accounts set up.
WARNING: as far as loans and charge accounts are concerned, forget it for at least a year! I tried to get a VISA card through my first new US bank with a local branch. They automatically check your credit rating through the major US rating agencies, and guess what? You have NO credit record, and hence no rating. I told them to check my Canadian records, and they said they would not even try that. So until some rating can be created over a space of time, nobody will talk to you about credit in any form. Your friendly neighbourhood bank at work!
I found a VISA card solution a different way. It happens I have been dealing with the Royal Bank in Canada for years. They have a subsidiary in the USA called RBC Centura, mainly operating in the south-eastern USA for "snowbirds". I opened a chequing account with them and asked them to issue a VISA card based on their knowledge of my records with the Canadian Royal Bank. Sure, no problem, and it worked. It even worked out well for on-line access. I had been using on-line access to my Royal Bank accounts in Canada for years. I set up access to the new RBC Centura account by computer, and then their systems allowed me to get access to both sets of accounts with one sign-in. They even allowed me easy transfers of money between Canadian and US branches of Royal Bank.
By the way, bill payments and bank-to-bank transfers in the USA are not as easy as in Canada - there are so many independent US banks that they don't co-operate well, at all.