I strongly feel that laws preventing the free movement of people are almost always negative. That's the most politic way I can think of to give my opinion.
Some details: I believe that anyone* who wants to enter this country should be able to do so, freely and at any time. Come to the gate, get your passport stamped, enjoy your trip. I believe they should be able to stay as long as they want and engage whatever legal activity they want while they are here. I believe that we have placed very restrictive immigration laws in place (going back 150 years, not just recently) so that if someone bypasses them to get here, we can smugly call them "an illegal" and deport them without regard to whether or not we should allow them to stay. I believe that politicians latch on to ignorance and a twisted us-vs-them-ism to pass laws that make current citizens feel good about stuff despite that they harm everyone. Note that none of this has to do with race, racism, or racial profiling.
But for those that don't want to come to the gate, don't want to get in legally, don't want to pay taxes and don't want to contribute to society, but want to ask for service's help, medical, school and resources, all without contributing, what do you do with them? that's the reason for the bill. Not that there are too many legal, hardworking immigrates, but that there are too many that don't want to do it the right, legal, proper and expensive way and instead what to take what they can. Even Mexico has limits on people/visitors/immigrants for how long they can get, the resources they can have when they are here, etc. So why can they have limits but not us? Nations have very little money to help people that do contribute, why continue to help those that don't?
Finally, specific to this AZ law, I think it's unconscionable to require residents to show papers to prove their citizenship, and it comes dangerously close (not just long-term slippery slope hypothetical close) to tyranny. This law was the first time in my adult life I can remember being embarassed to be an American. For all my disagreements with other things like the Iraq war, certain social program, etc., they never actually caused me to feel shame like this. If in the long-term things like this become accepted and commonplace, I will seriously consider becoming an ex-pat.
That's going to be the issue, but if they ask you to show, in the process of stopping you for something else, then that's a different topic. Stopping them outright for a check is wrong, checking them while they are speeding or having other issues, well why not. If they have a passport/visa then fine, life moves on. If not, they go in the system.
You say you are embarrassed to be an American, yet this is a State's choice. You say you are for less Fed, more states right, this is a choice that 70% of the states citizens wanted. It might not be something you like, but it seems to be spurring something that a lot of citizens want. that's a big difference and a far cry from bills/laws that the fed makes that citizens have no choice or control in.
* yes yes, except the terrorists who hate America. FFS.