To pull all the comments together, I'll take the question in parts.
I usually state something in the Readme which reads like "Code written by [name and alias], licensed under GPLv3" ... Is this sufficient?
In the sense that anyone who wants to exercise their free-software rights can safely do so, yes. But it's very helpful if you do a little more, as vonbrand says.
do I need to include the full license on GitHub?
If you're publishing a derivative of someone else's GPL software, you are required to do so (GPLv3 ss 4, 5). If you're the original author of the entire work then the GPL doesn't bind you, even though you're publishing under it, but it's still very much best-practice to include a copy of the licence.
GitHub has a per-repository "licence" field, and it's helpful for people automatically searching for free software if you set this correctly also.
Furthermore, it's very helpful if the header of every file in your codebase includes a clear copyright statement, and says what licence it is distributed under.
The GPL's full name is "the GNU General Public License", by the way, and as Brandin points out there is best-practice guidance at the bottom of the licence on how to apply it to your program.
And for other licenses, such as MIT?
Given that at least some authorities find the term "MIT licence" ambiguous it would be an equally good idea (to maximise user convenience, and minimise ambiguity) to include a copy of any other licence you intend to publish under.