Travelling with a firearm (in general)
I've actually seen this scenario, where there has been an assumption by the (US) traveller that they would just be able to enter another country with their firearm.
However, as @Mark Mayo states, each country has their own laws in this regard. It's important to realise other countries are likely to have much stricter rules.
Typically, as long as you follow the rules of your local airport / carrier, you'll find out about local restrictions without too much issue. This means declaring that you have your handgun at the earliest opportunity (at border control / customs etc.) on departure and entry. However, it's much wiser to find this out in advance (or you risk having your property confiscated - and / or potentially face criminal prosecution).
For example, if you travel to Canada with your handgun, as long as you declare it at border control, they'll usually just inform you that you can't bring it in without a licence - and often give you the option of not travelling to Canada or allow you into Canada without your gun (I believe they retain it until your return).
Similarly, the UK requires a licence (in the form of a visitor's permit) to bring a gun into the UK. However, that gun has to conform to UK rules - your handgun would be a prohibited weapon (banned) and you won't have a UK legal reason to possess any gun (self-defence).
If you declare it at customs, they won't allow it in the UK - but you're unlikely to be prosecuted if you've been open and honest about what you're carrying. Disguise that fact (as @Thorsten S. says) and you're likely to be arrested and imprisoned - illegal possession does carry a 5 year minimum sentence.
Concealed weapons and their use
First, I'd probably reconsider where you intend to travel to. If you're going on holiday, I'd imagine it'd be much more relaxing and enjoyable to go somewhere where you don't feel the need to be armed. If you're travelling on business, your employer is generally expected to provide appropriate security for any "high-risk" nations.
If you find a country where you can bring a your firearm (which is likely to be rare for purposes other than sport and hunting), there will probably also be strict rules about how and where you carry and use it.
Every place you visit and each building you enter may have different rules to what you're used to - really, border entry would be the least of your concerns. It really would be a minefield of situations where you could land yourself in trouble (either with the authorities or someone misinterpreting your intentions). I'd strongly suggest that you don't place yourself at risk in this way.
Very broadly, you shouldn't take anything to another country unless you are fully aware of its legal status there.
Obviously not an exhaustive list - but key topics are:
- Weapons (knives, firearms etc.).
- Chemicals (including explosives).
- Fresh food.
- Plants and animals.
- Money (in larger quantities, the actual amount will vary - often by currency too) or "cash equivalents" such as precious metals/stones, market-holdings and bonds.
- Sensitive information (consider religious states and kingdoms with lèse majesté laws), religious writings and pornography.
- Drugs (legal or otherwise).
It's not a coincidence that most countries ask you to sign a declaration, mentioning the above list (in various forms) on entry.
Being a citizen of another country does not grant you any special exemptions over local laws in the country you're visiting.