Only the First Amendment explicitly states it is enumerating rights that "Congress shall make no law respecting"; the rest simply refer to rights of "the people," "any person," or "the accused." Thus it seems to me it's entirely legitimate to assume that the entire Bill of Rights, with the exception of the First Amendment, describes rights that all U.S. citizens have and that neither the federal nor state or local governments can take away. (Lucky for us, the First Amendment very definitely was part of this mysterious incorporation process that somehow the Second Amendment was left out of, because it apparently seemed pretty obvious to everyone else that if Congress can't censor what you say, neither should your state legislature or city council.)
In general, I think the rights conferred upon citizens by the federal government should be the absolute minimum recognized by states, that states could grant extra rights but couldn't take away any already granted by the federal government. How can a state claim jurisdiction over something that the country of which it is a member has already granted to every citizen (resident?)? If people could be citizens of the U.S. without somehow being under the jurisdiction of any other governmental body, there might be a case to be made that states can then do whatever they want and if you don't like it you can go someplace else, but to me it seems like it's entirely pointless to be granted a right by the federal government if it's entirely possible every single state (you have to live somewhere!) would curtail it. Then where would you be?
Thus, the first issue I have with state's rights: that (in some cases) a person who is a resident of both the United States and a specific state is thought to somehow be primarily under the second-tier state's jurisdiction and only secondarily under the jurisdiction of the first-tier, overarching federal government. (I know the Ninth Amendment confers all powers not enumerated in the Constitution to the states, but I'm talking here about the ones I think it does very definitely enumerate.)
At first blush it seems all very nice and reasonable that local governments are best situated to rule and regulate themselves, that the people in an area may have different concerns than the country overall and that they might prefer to regulate different aspects of life or regulate life in different or opposing ways to those of some other locale. After all, who wants some distant bureaucrat telling them how to best run their own affairs? However, I think local rule can only be truly fair and effective when communities are remarkably homogeneous and when everyone there chose to live there.
Regarding the inevitably incomplete homogeneity: I think local rule tends to silence minorities (of all varieties) even more effectively than federal rule does. On a nationwide scale, people who are the minority in a city or state might be either a much larger minority (say, African-Americans in New Hampshire versus the entire U.S.) or even the majority (say, conservatives in Massachusetts versus the country at large). There is simply more room for a meaningful debate about equality and representativeness and more impetus toward protection of minority rights on a nationwide level due to the increased visibility and larger numbers. Protecting the rights of one gay couple in some random small, overwhelmingly conservative and/or religious town may not seem like a legitimate issue to the rest of the townspeople (it's just a few people being deviant or provocative or asking for special treatment), whereas on the national stage gay rights are at least a recognized concern, however divided governmental bodies and the populace at large may be on the issue. Minorities deserve protection no matter where they are, but if they're a small enough minority, a lot of times it's not even clear to the majority that they are a distinct minority that merits protection. It's helpful for a more removed federal government to make these calls.
Regarding choice: If every state had a clearly-enumerated list of laws that was easily compared to that of every other state and if all residents of a state chose to move there based on their willingness to be bound by those laws, I would have no issue whatsoever with states enacting whatever laws they so desired. However, the vast majority of people (myself excluded) probably did not simply pick a state they wanted to live in based on the political and social attitudes represented therein. People live places for all sorts of reasons: they were born there, their parents moved them there when they were still minors, that's where they went to school and so they stuck around, that's where the job they found is or their job relocated them to, that's where their or their spouse's family is, that's the kind of climate they enjoy or has the sort of landscape they find enjoyable (hiking, rafting, skiing, beaches, etc.). While many of these may in some way reflect legal structures (regarding industry regulation or educational funding or what have you), the laws of a place are very rarely a conscious consideration of any one individual when planning to relocate. It could be argued that that is the fault of the individual, not the state, but I find this pretty unrealistic.
In addition, there are plenty of reasons why the state you're in now is likely to be the state you stay in. Many poor people lack the resources to relocate or the skills to find employment in certain other regions of the country. And, more than anything else, people just tend to go for the status quo, so all other things being equal, they tend to stay put.
Moreover, it's not like everyone is just born into some limbo state and brought to the age of majority in an entirely neutral place then left to select where they want to live. I think it is deeply unfair to have minors be stuck some place that may have pretty extreme laws they never voted for or even tacitly accepted by willingly choosing to reside there. Pity the poor gay kid stuck against his or her will in a red state without hate crime legislation, gay marriage, civil unions, nondiscrimination policies that include homosexuality, etc. What about teenagers who want contraception who live somewhere they can't get it without parental consent? What about atheists (or just science nerds) living somewhere that mandates equal time for intelligent design as evolution? What are they supposed to do? It's hardly their choice to be there, and they have absolutely no representation. (I'm sure equally terrifying examples could be generated regarding living in a ridiculously liberal state, but I'm having trouble coming up with those since I just don't think like that. Or, you know, maybe not, since liberal states tend not to restrict what people can do; you can always be more conservative than your surroundings, but being more liberal than your surroundings often runs you up against legal bounds.) Simply because the adult population of where they live is unlike them, these children are stuck in what is quite possibly a hellish existence for them. With a stronger central government making many of the decisions that are now left up to the states, laws would probably be a little more balanced and mainstream, severely disturbing a much smaller percentage of the populace.
(Of course, the same argument can be made regarding citizenship on the national level—a relatively small percentage of the world's population actively chooses in which country they want to live, and even if there were free movement between all nations, minors would still be stuck where their parents want them to be. This is one of the reasons I think Hobbes's social contract is full of crap—nobody alive today actually made a conscious choice to leave the state of nature and join into the leviathan. Regardless, we as a country can't really do much to change that, but we could try to make life suck less for kids who aren't like their parents who happen to live in this country.)
And now the flip side of mobility: especially nowadays, everyone moves across state lines. (OK, not everyone, but 22 million people moved across state lines between 1995 and 2000, which appears to be about 13% of the population. In five years. So obviously way more than that has moved across state lines at some point in their lives.) It is absolutely impossible to be aware of every difference in law between your old state and your new state, especially before moving. (I'm still blown away that there's not a nice handy website that encompasses all state laws ever and permits them to be compared; states' websites are hit or miss, and they're all obviously in different formats and pretty much impossible to compare.) So you move to another state because of your job or something, and then the next thing you know, you learn you can't talk on your cell phone while driving, or your 17-year-old son can be charged with statutory rape for sleeping with his 17-year-old girlfriend, or you can't get a no-fault divorce without a year's separation first, or gay partners of biological parents don't have any legal rights over kids that were previously considered equally theirs, or it's illegal for medical insurance to cover abortions (or the reverse of any of these scenarios, though again, I'm having trouble seeing how discovering that the opposite of any of these was the case would be as problematic—I guess there are plenty of people who would like the 17-year-old boyfriend of their daughters to be made sex offenders as punishment for premarital sex (essentially) and would be disturbed by the kind of laws that doesn't make it statutory rape if both parties are under 18 and within two years of each other's ages or whatever, right?). Most of these situations are ones that people would never expect to find themselves in in advance (divorce, abortion, children committing sex crimes, etc.) so they aren't that particular about seeing what the law has to say until it's suddenly applicable. How do you know, short of reading the entire legal code (which I assume is pretty obviously ridiculous) what random laws there may be that might crop up to impact your life in ways you never imagined?
If laws were consistent nationwide, there might still be things you never thought to look into, but you'd have a much better chance of hearing or seeing something about them if they were applicable to everyone in the entire country. You'd know that if it only took a month for your brother and sister-in-law to get a divorce, you could get one in about the same timeframe without having to remember that they live somewhere that's not New York (although actually I think that changed recently). You'd know if you read some story about surrogate mothers winning custody of the kids they bore that you'd better beware of that sort of thing if you're using any sort of nontraditional means of reproduction. (You'd know if you read some story about surrogate mothers suing for and losing custody of the kids they bore that it obviously wouldn't be an issue for you, rather than getting lulled into a false sense of security and then, ta-da, oh, that story took place in some other state and is actually the exact opposite of what would happen in your state.)
Basically I just don't think modern life works well with such a fractured system. In the age of the internet (which brings information about myriad such situations people find themselves in, often without making it clear how this may differ from state to state) and of frequent inter-state relocations (which suddenly and entirely (and largely unnoticedly) change the rules by which one is to abide), so many independent sets of laws seems pretty ridiculous. And then of course there are the ones, like most family laws, that turn into a giant mess when one party relocates. If you have a child with a partner you're not married to and then that partner leaves you and takes the kid to another state, which state's family laws apply? If you're gay and the state they went to doesn't recognize civil unions, domestic partnerships, or non-biological parents' rights but the state you resided in and formed your family in did, now what? Wouldn't it just be easier if it were the same everywhere? So yeah, I think states are pretty anachronistic, personally.