I understand that the first amendment enshrines freedom of speech. However, can the location where someone speaks pose a limit to this? Is it different between public and private property?
I'm going to have to disagree with people saying that location is irrelevant. In some cases, the government does have enhanced power to regulate speech on certain kinds of property, because the government-as-landlord can do things the government can't do in other contexts (just like the government-as-employer has more leeway to regulate speech of employees). The relevant question is whether or not the location is a public forum, which means whether or not it's an area that's open to general discussion and debate. Nonpublic forums include military installations, courtrooms, government office buildings, post offices, etc. None of these places are set up for general public discussion. They all have specific purposes, and are allowed to be regulated to ensure they meet those purposes.
If you are in a nonpublic forum, the government still cannot generally discriminate based on viewpoints (there could be some military-related exceptions when civilians are subject to military law, but the general rule is no viewpoint discrimination). However, it can establish extensive rules to regulate permissible content of speech, and is only bound by the requirement that the rules be reasonable. There is a limit (the Board of Commissioners of LAX passed an ordinance to ban, and I quote, "all First Amendment activities" in the terminal; this was not considered reasonable), but the fact that the government is bound by the First Amendment doesn't mean they have to abandon any authority to control the use of their property.
However, not all government property can be a nonpublic forum. The government doesn't get to designate an area traditionally used for public discussion (like a street, a park, or similar) as a nonpublic forum; it must treat those as public forums. (There's an intermediate level too, which is designated as a public forum for certain purposes but is not a traditional public forum, where the rules are intermediate between public and nonpublic forums). In a public forum, all the normal rules apply; the government can't ban political speech on a sidewalk any more than it could ban it in on your front lawn. The government can impose reasonable time, place, and manner restrictions as long as they're narrowly drawn to advance an important government interest and they still leave plenty of alternative ways to speak, but it doesn't get the deference on content restrictions seen in nonpublic forums. Content-based restrictions in public forums have to meet the same kind of standards needed for the government to impose content-based restrictions on private property.
Which brings us to private property. This is generally classed as a nonpublic forum under the federal Constitution, although some states (like California) class some private property (like shopping centers, which are a modern gathering place equivalent to traditional public forums like town squares) as public forums. Also, under the federal Constitution, in a privately owned company town the traditional public forums are still public forums despite being privately owned. What the distinction means here is whether the trespass laws can be used by the owner to restrict speech. In public forum private property, you cannot be convicted of trespass for engaging in what would be protected speech on a government-owned street.
Other than that, the government is as restricted in regulating private property as it is in regulating public forums, if not more so. The property owner, however, can regulate it, and the state backs them up through trespass laws.
You are confusing the right of freedom of speech with other concepts that can be linked to speech.
Let me give an example. In the U.S. you have the right to say that your next door neighbor has sex with rodents. However, if your your statement is false, you are liable for the tort of slander. You are free to make your slander but you have to pay for any injury you may cause if exercising your free speech interferes with the legal right of another.
This is unlike most countries where you can be jailed or enjoined for making false statements.
The same goes for the location of your free speech. Your right of free speech does not include the right to engage in trespass.
Edit: Unalienable, not inalienable
It doesn't end, regardless of location or circumstances. It's "unalienable", which means "unable to be taken away from or given away by the possessor".
The First Amendment, along with the other nine in the Bill of Rights are considered unalienable. They can be suppressed by oppressive governments, but the Founding Fathers understood that they can never truly be taken away.
You can always speak your mind, even if you're gagged and locked in a room by yourself, or standing in front of a firing squad.
You can always bear arms, because any object can be a weapon, even your own body.
There are some rights listed that are more subtle, like the 4th Amendment, but if they're violated regularly, backlash from the community will be an inevitable result at some point.
The 10th Amendment, also follows this logic, once you consider it. People decide what their rights are, these are never truly "given" by governments. People may surrender their rights for a time, but eventually they will reclaim them. With this understood, it's obvious that additional rights would be declared (not "given") at the state level, and finally the ultimate arbiters, the people themselves.