There's no one right answer to this question. What do you want to do with the photos? What is your reason for wanting to sort or tag them? The answers to those questions might help guide your decision on which approach to take.

Here are some Pros and Cons of different approaches:

Tagging

Pros: Easily find things by tag, even across different categories. Want to find a picture of a wiener dog in a party hat, but can't remember where you put it? Just search for the tags "wiener", "dog", and "hat". Some tags can be added automatically by various applications out there. Images can contain multiple tags.

Cons: Have to add most useful tags yourself. That means looking at each photo and thinking about all the tags that apply and writing them all down. There are apps that can help, but it's still a lot of work. Also, having all your photos in a single directory is likely to slow down your computer every time you open that directory.

Using the File System

Pros: Very easy to create new folders, can have whatever structure you want and put things anywhere you want. Automatically adds some metadata like creation (or at least download) date, and possibly website it came from.

Cons: Sorting is time consuming. Each photo can only be in 1 folder, unless you make copies, which is a mess. Doesn't contain any other metadata besides what the OS puts there.

Using a Photo Management App

Pros: Gives you several ways to slice and dice your collection, including making projects, folders, slideshows, etc. Can put all your photos into a single repository and can have a single copy of each photo, but include them in multiple collections. Can also tag them and find them based on tag.

Cons: You are probably stuck with the app's management system in some ways. It can be difficult to have multiple drives containing photos that you might want in the same project, etc. You still need to do the work to use the app's tagging system.

I'll be honest, for me as a fairly experienced Photoshop user, taking and organizing my own photos was a mess until I started using iPhoto. Even though it wasn't a very powerful app in terms of image processing and the like, having it automatically sort my photos by event and allowing me to create my own collections was incredibly useful. I thought at first that I'd hate having all my photos inside the document bundle where I couldn't easily find them, but it turns out not to matter nearly as much to me as being able to organize my photos. I later moved on to more sophisticated software but it still has very good organization tools.