There's nothing wrong with experimenting with, dating, or even marrying a cousin. There are some countries and a little over half of US states where the bigotry against marriage equality extends to preventing first cousins from marrying, but there are many places where marrying a first cousin is legal and common. I'm only aware of a few US states where sex between first cousins is technically illegal, so check the laws of your state if you are concerned. It should be searchable on your official state website.
There is no rational reason for keeping laws or taboos against consanguineous marriage. Personal disgust or religion is only a reason why one person would not want to marry their cousin, not why someone else shouldn't do it. An adult should be free to share love, sex, residence, and marriage with ANY consenting adults. Youthful experimentation between close relatives close in age is not uncommon, and there are more people than you'd think out there who are in lifelong healthy, happy relationships with a close relative. It isn't for everyone, but we're not all going to want to have each others' love lives, now are we?
Some people try to justify their prejudice against consanguineous sex and marriage by being part-time eugenicists and saying that such relationships inevitably lead to “mutant” or “deformed” babies. This argument can be refuted on several fronts. 1. Some consanguineous relationships involve only people of the same gender. 2. Not all mixed-gender relationships birth biological children. 3. Most births to consanguineous parents do not produce children with significant birth defects or other genetic problems; while births to other parents do sometimes have birth defects. 4. We don’t prevent other people from marrying or deny them their reproductive rights based on increased odds of passing along a genetic problem or inherited disease. It is true that in general, children born to consanguineous parents have an increased chance of these problems than those born to nonconsanguineous parents, but the odds are still minimal. Unless someone is willing to deny reproductive rights and medical privacy to others and force everyone to take genetic tests and bar carriers and the congenitally disabled and women over 35 from having children, then equal protection principles prevent this from being a justification to bar this freedom of association and freedom to marry.
Some say "Your cousin should not be your lover." That is not a reason. It begs the question. Many people have many relationships that have more than one aspect. Some women say their sister is their best friend. Why can’t their cousin be a wife, too?
Some say “There are so many people outside of your family." There are plenty of people within one’s own race, too, but that is no reason to ban interracial marriage. So, this isn't a good reason either. Let consenting adults love each other the way they want!
No, incest should not be permissible between first cousins because it is wrong and because we no longer live in a time and age to where that can be seen as morally right or acceptable. Incest is wrong because you are being intimate with someone you are related to and mostly someone you share the same bloodline with.
While it was something that was practiced in the European aristocracy, I think it's not advisable. The fact that being so closely related increases the chances that their offspring will have genetic defects. Also, such a close relationship would be confusing for any such offspring or perhaps, more correctly, it would come with a lot of social stigma for said offspring.
No, I don't think first cousins should be aloud to marry. Second generation and beyond, may be permissible under certain circumstances. The act of propagating with one's own cousin is something that has been shunned for many years with good reason. Though in respecting the rights of an individual my stance remains soft, with a bit of a lean towards yes on this question under certain circumstances. In cases of adoption or other individual circumstances that arise on a case by case basis, a blanket statement like, "Should incest between first cousins me permissible" does not leave room for situations that may actually make this old taboo null and void.