There are many different ways to go about learning web design. Everyone learns things best in different ways. School may be the answer for some people. For the rest of us a strict program that offers no flexibility leads to nothing but cramming without retention! In most computer related fields the technology is changing so quickly that it is impossible for curriculum programs to stay up to date. My recommended method involves a combination of books, internet and trial and error.
The first thing I would do is get a few books on the subject. These books will provide a great introduction to web design and later on they will become a quite handy reference. I recommend Head First HTML and CSS Mastery. Head First does an excellent job teaching you all the basics of html plus a little about the web in general. They market themselves as "brain-friendly" and use a lot of learning techniques (i.e. puzzles, exercises and funny stories) to help you understand and retain the information in the book. CSS mastery is for a slightly more advanced level and really does not need to be read cover to cover. I would simply skim through it after reading Head First so you can get an idea of what sort of CSS topics it covers. This will allow you to more quickly be able to reference the book later on.
Once you have gotten everything you can out of these two books it's time to start designing! Ultimately this is the best way to learn...Through trial and error (and believe me there will be plenty of errors along the way!) One of the best ways to start designing is by going on one of the design contest or freelance design sites out there. There are quite a number of these and each one has its pros and cons. I recommend 99designs.com simply because they don't charge you to search through and submit designs, and they have a large amount of web design projects listed. There is also a way for you to see what other designers have submitted on the project so it might help you get a starting point more easily the first few times. The best thing about 99designs for a beginner is that you can see what the design project is and get practice trying to design for the project. If you don't like your design, you don't have to submit anything, but if you do like your design, you can enter it in the contest and possibly win some money! The biggest flaw with 99designs is that they don't have an area for "coded designs" which you should know means designed with html and css. In order to get the practice you need design the project coded. However if you decide you want to submit your design you need to find out what format the client would like (i.e. PSD or coded). If it's PSD it's time for you to sharpen your photoshop skills!
Almost every web designer needs to have some familiarity with Adobe Photoshop and Illustrator (these are the industry standard graphic design programs). This will allow you to add illustrations and images to your site to help set it apart from others! Unfortunately these are both expensive programs for the beginner. If you're looking for a deal you might want to check Ebay for a slightly earlier version. If you decide to get it on Ebay make sure you get all the license codes and read all the fine print from the seller before buying it. You should also be aware that there are free or open source design programs out there as well. These include Gimp and Inkscape. If you can't afford the Adobe products at the moment these may help tide you over so you can continue to sharpen your web design skills.
There are lots and lots of books out there that teach you the basics of Photoshop and Illustrator, but I have yet to find any that I feel are worth recommending. If you come across any that you think are really awesome let me know :). I have instead found the best way to learn these design programs is to simply start designing! There are hundreds of great resources online that give you tutorials or answer questions you might have about how to do something in photoshop and illustrator. Youtube is also a good source for video tutorials. Usually the first few times you try to design in these programs you feel overwhelmed. If this is the case for you then I recommend spending a little while going step by step through some tutorials. Once you have done this a few times you should feel far more confident to begin designing on your own.
My final recommendation (after you have done several practice coded designs) is to do a design for yourself or a loved one. By doing this you will learn about hosting, domain names, getting sites indexed and lots of other very important intricacies that you as a web designer should know!
One final note...try not to ever feel defeated! Being self taught can be challenging and others may not agree with your methods. The "norm" for so many people is to go to school and get a degree. That approach is great for some people. For the rest of us why put yourself through something that doesn't work for you and go broke in the process when there are millions of great resources out there. If you truly want to be a designer without the schooling then go for it!