Let's look at the Second Amendment:
"A well-regulated militia, being necessary to the security of a free state, the right of the people to keep and bear arms, shall not be infringed."
I believe that this text means exactly what it says: that the right of the people to keep and bear arms shall not be infringed. That means that the people - as in the First and Fourth Amendments, the citizenry of the United States as a whole, and as individuals - have the right to arm themselves as they see fit, and to carry those arms with them in their lawful pursuits.
Many folks point to the words "well-regulated militia" and say "the National Guard is all that means!". There are three problems with this:
1) "Well-regulated", when the Bill of Rights was drafted, did not carry the meaning it does today of "controlled by specific written rules with the force of law". That meaning of "regulation" is definitely a 20th-century creation, and quite probably post-1950. In the late 1700s, that term meant "well-equipped and well-trained".
2) "Militia" means "everyone able to hold and use a gun". (Yes, the Founding Fathers limited that to men, but I would advocate opening it up to every citizen, male or female.) This is not just supported by the words of the Founding Fathers, but is written into current US law: every male between 17 and 45 who is, or has declared his intention to become, a US citizen is a member of the militia. See 10 USC 311.
3) A basic legal principle is that a term means the same thing wherever it is used in a document. "The people", as used in the First and Fourth Amendments, clearly recognizes an individual right; for it to recognize a collective right in the Second Amendment violates that principle. Conversely, for it to convey a collective right in the First Amendment would recognize the right of the government to have freedom of speech, the press, and religion, and the Fourth Amendment would recognize the right of the government to be free of unreasonable search and seizure by the government - both, plainly silly.
Another argument from those who would deny the plain meaning of the Second Amendment is "but that would allow anyone to have nuclear weapons!". To which I reply: Yeah...so what? Using them would still be murder except in very unusual circumstances. In the early days of the US, the Second Amendment was understood to allow anyone to have any military weapon they could afford and use. There are several records of privately-owned warships and cannon from those days.
The Supreme Court has also recognized this argument: in their ruling upholding the law banning sawed-off shotguns in the 1930s, they held that, since they were not weapons of military utility, they did not fall under the Second Amendment's protection. (They held so incorrectly: sawed-off shotguns were a favorite of troops fighting close-quarters actions, even through Vietnam, and, I believe, are still in the US military inventory. The reason they ruled as they did was that the plaintiff had died between the filing of his appeal and the oral arguments, and nobody argued for the appeal.)
Basic rights are often problematical. I will certainly grant that there are people that feel nervous that their neighbor might be armed. Many of them were quite vociferous in this during the debate over the Minnesota Personal Protection Act, and in the wake of its passage. They conveniently ignored that the bad guys - criminals who, by definition, ignore the law - were already armed, and the only people who would go through the training and permitting process would be the law-abiding citizens who would otherwise go unarmed. Those are not the people one need worry about.
I'm not the first to travel this path. Professor Sanford Levinson of the University of Texas School of Law, a leading legal light of the ACLU who is by his own admission no friend of the Second Amendment, nevertheless concluded that it does mean exactly what it says, and recognizes an individual right. His paper on the subject, The Embarrassing Second Amendment, should be required reading for anyone wading into the discussion.