As for web typography, you want to check of the latest generation of font embedding because that's where the real magic's at. The most known web font solutions are: CSS3 (font-face), typeface.js, cufon, flir, sifr, Google Font API, Typekit.

Here's a list of quality books on design and typography:

    A History of Graphic Design
    Before & After Page Design
    Designing Brand Identity
    Emotional Design: Why We Love (or Hate) Everyday Things
    Graphic Design Solutions
    How Designers Think
    Information Dashboard Design: The Effective Visual Communication of Data
    Making and Breaking the Grid: A Graphic Design Layout Workshop 
    The Complete Manual of Typography
    The Designer's Guide to Color Combinations
    The Elements of Graphic Design: Space, Unity, Page Architecture, and Type
    The Elements of Typographic Style
    The New Typography
    The Visual Display of Quantitative Information
    Thinking with Type: A Primer for Designers
    Typography Workbook: A Real-World Guide to Using Type in Graphic Design 
	

    About Face 3: The Essentials of Interaction Design
    Ambient Findability: What We Find Changes Who We Become  
    Bulletproof Web Design
    Cascading Style Sheets: The Definitive Guide
    Designing Web Usability
    Designing With Web Standards
    Designing the Obvious: A Common Sense Approach to Web Application Design
    Designing the User Interface: Strategies for Effective Human-Computer Interaction
    Don't Make Me Think: A Common Sense Approach to Web Usability
    Homepage Usability: 50 Websites Deconstructed
    Information Architecture for the World Wide Web: Designing Large-Scale Web Sites
    Information Architecture: Blueprints for the Web  
    Interaction Design: Beyond Human-Computer Interaction
    Rocket Surgery Made Easy: The Do-It-Yourself Guide to Finding and Fixing Usability Problems
    The Essential Guide to User Interface Design
    The Inmates Are Running the Asylum
    Thoughtful Interaction Design: A Design Perspective on Information Technology
    Web Form Design: Filling in the Blanks

    Before & After Page Design
    Typography Workbook: A Real-World Guide to Using Type in Graphic Design
    Thinking with Type: A Primer for Designers
    How Designers Think
    Information Dashboard Design: The Effective Visual Communication of Data
    Making and Breaking the Grid: A Graphic Design Layout Workshop
    The Designer's Guide to Color Combinations
    Emotional Design: Why We Love (or Hate) Everyday Things
    The New Typography

This was a great exercise as it's something you can do as you walk about, or in your case, you can apply it to websites in much the same way. It gets you in the right frame of mind. By constantly being on the lookout, it starts to become ingrained, and you can start to identify fonts automatically. You judge what you like and dislike and what you would change to achieve a desired effect.

As with many things, critical constructive thinking is the key. And perseverance.

Then go and get a Typekit account http://typekit.com/ and start experimenting.

Good design (any kind of design, not just web design) requires time and iteration. It's a skill, you have to practise it.

for long term understanding, it's just like any other skill/trade/art: keep researching, learning, searching, finding inspiration, figuring out what you don't know, so you know what you want to learn next, and if it's worth learning.

you can "start hacking" by mimicking (copying) certain styles, seeing how they work, analyzing their advantages and disadvantages in different situations (just like different programming languages)... etc.

I think you're already way ahead of the crowd just by knowing how important typography is to the successful design of site, best of luck.

Or, just use Helvetica with big sizes and good contrast.

If there's one seminal book - that would be The Elements of Typographic Style: http://www.amazon.com/Elements-Typographic-Style-Robert-Brin...

That's a deep dive into it though. If you're looking for a shortcut you can get a quick hold of them here: http://www.amazon.com/Fundamentals-Typography-Gavin-Ambrose/...

At Future Simple (http://www.futuresimple.com) we think about that a lot, sometimes agonizing over it. We still have a lot of work to do :) - as any designer will tell you - we are never satisfied with our work...

But I agree with aba_sababa - you can learn specific technical details about fonts and typography in books, but an overall sense for good design has to be assimilated over time. So, when you see sites that you like, use Firebug and copy their font settings. Or if you see a site that has crappy typography (eg. Hackernews) then use firebug and try to fix it.

http://www-cs-faculty.stanford.edu/~uno/abcde.html

I'd recommend checking out OpenFontLibrary.org & Kernest.com (shameless plug) for you can download the fonts, use them, and understand how they work in all the work you do - not just web browsers.

Typeface Classifications: http://typedia.com/learn/only/typeface-classifications/

But start with this one: "The Non-Designer's Design Book" -- it'll teach you how much of "great design" is "great typography".

check out this ebook