If you have a computer science degree there are plenty of entry level openings for web development. I would recommend actively applying to some of those and seeing what you can come up with. The best way to learn this stuff is really to get thrown in the deep end. With a computer science degree it probably won't be all that difficult to pick up considering the first language is always the toughest and you should already have a basic understanding of programming from a fundamentals stand point.
The only question to really ask yourself is what technology you would like to work in. The top ones are PHP, .NET, Ruby, Python and Java listed in no particular order. In most cases a company will only deal in one or two of those. Something like .NET might be easier to pick up if you already familiar with C++ or C#. I wouldn't expect a good college engineering program to go over much web specific stuff. Web development is really a speciality of software engineering like embedded systems or something.
Most people looking at your resume for an entry level position are just be concerned that you understand programming from a fundamental aspect with the degree. You probably would have the easiest time finding something at a company specializing in .NET though with just the degree. Finding a PHP position would likely be a little more difficult unless you learned the language yourself but still not impossible.
You will naturally pick up the client side stuff as you work on projects. It is pretty easy to pick up HTML and CSS. The tough part is mastering it. However, as an engineer you will probably only need to know the basics for the time being as a front-end developer will be most responsible for HTML and CSS stuff.
Without any type of internship in web development or prior experience I wouldn't expect an amazing salary of anything but average for an entry level engineer is probably around $40K – $50K. That is in the US… not sure about other places.