Yes, you can do design and development simultaneously. But generally companies take designers and developers seperately.
So if you want to go for development, the basics you are already doing, and the most common server side languages are ASP.NET, JSP, PHP + MySQL.
Now please don't ask which one is better from the three, as all of them is good and all of them has their own area.
PHP + MySQL being the open source is most common platform for development.
ASP.NET is generally used by enterprises. As it have much more cost incurred.
JSP is generally used by Banks, or other firms which needs strict security.
Besides this, you can also use Perl, Python using CGI.
Yes, you can use C++ too, but then C++ is generally not used for web development.
Others include, HTML, CSS, Javascript, jQuery etc. for Front End.
But in Web Development, you need to keep yourself updated with the latest technologies, means you can't stop learning after doing few courses.
These are the general web development side. Now if you want to be a RIA developer, you need to know Flash, Adobe AIR, Flex, HTML5, another platform is Silverlight.
So you see, learning doesn't gets end.
Best of luck