I have created a piece of software that can be sold internationally.
Do I need to register my logo/trademark nationally?
Intellectual Property Law Attorney
My colleagues have noted that you should register the copyright in your computer program. Good idea, you should do it [with the aid of an intellectual property attorney].
But I think you want to know how to protect the name and logo that you're using to brand your program. That is done by, first, performing a trademark search to determine if anyone is using the same or confusingly similar name or logo, and then, if no one is, by actually offering the computer program for sale under your chosen name and logo.
You can then register the trademark rights you've created through use of the name and logo with the US Trademark Office and with the other trademark offices throughout the world. There is a process that allows you to seek registration in more than one country at a time --- BUT, be warned, there is no "international trademark registration" [except for the European Union] and so each country where you seek registration will evaluate your application under its own laws.
I hope you take from this that protecting your brand is not best done by a novice but by someone who knows the ins and outs of the system. Hire a trademark attorney. Good luck.
Patent Application Attorney
Software is protected by in the USA by Patents and by Copyright laws.
With few exceptions, Internationally software is protected by Copyright laws only.
International treaties allow you to register your copyright in the USA and be valid evidence almost worldwide.
A trademark protects the brand that you intend to use to commercialize the product. Since you intend to commercialize it internationally, you should register your trademark in every country where you intend to sell it.
I agree with my colleague -you should both copyright your software (please see the link below) and trademark it. Hire an IP lawyer for help.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.