There are two web problems I'd like to see solved or solved in a better way:
a) Discovering new web sites. In the early days it was What's New | What's Cool from Mosaic and then Netscape. Yahoo still has a pretty cool What's New web site, and modern day What's New/Cool sites include Wikipedia front page, Digg.com, and Del.icio.us. Another method of discovering new sites is through blogs, bloggers put up links they find interesting. People don't really "discover" or look to discover new sites through search engines, they're more looking for information. And finally, people discover web sites offline through advertising, word of mouth through friends and family, and television.
b) Filtering of web sites/information. The other problem (and I believe fairly major) with the Web is too much information in general, and too much information within a given site. It's certainly hard to control what a site gives you, I have a hard time filtering and finding the information on my own site. With the explosion of blogs, attempts to consume all that content has come in the form of blog readers. Other tools in the past include bookmark managers, alerts (like CNN or Google alerts), "My" pages, and blog sites themselves. Tabbed browsing helps users consume more sites and popup blocking helps to filter out bad content.
More on these topics later.