Can someone explain to me the "real world" differences between a 2 way and a 3 way speaker...?

It'* actually english...
1 way = full range speaker(Full range driver)
2 way = woofer mid/or high(2 actual speakers)
3 way = woofer mid AND high(3 actual speakers)

Here'* some good info...
Loudspeaker - Wikipedia, the free encyclopedia

In terms of "english"... a full range or 1 way speaker produces some of the audio spectrum, not all of it. So, you miss out on all of the real bass punch of drums and the high twang of guitars.

With 2way and 3 way speakers, each individual speaker has its own tasks. High frequency, MID and LOW frequencies.

The number of speakers. 2 way typically have a tweeter and a woofer. 3 way will come with a tweeter, mid driver, and woofer.

So... a 3 way is better than a 2 way...?

The best solution would be three separate speakers appropriately placed in the vehicle. Tweeters handling the highs, mid range drivers handling the mids, and a subwoofer handling the bass.

In your case, I would not recommend a three way speaker. I would recommend a component set with tweeters and mid-drivers for the front doors and either 6x9s in the rear handling bass or a sub in the trunk. Rear doors speakers are optional...

I am looking at:

6.5" for front doors - not component - just 6.5 and leave the factory tweeters in the upper door as they are.

I can get a two way or a three way 6.5".

Then 6x9 in the rear deck - I can get either a two way or a three way.

Then the 5.25" in the rear doors - I can either get a two or three way.

So, what is the "better" set-up - two ways or three ways?

Without bypassing the amp you are going to only get mid range to the driver doors, so anything other than the mid range speaker will be rather pointless. If you must choose between a 2 or 3 way, I would go with the 2 way in this situation.

Same goes for the rear doors, only mid range is sent if you do not bypass the amp so again if you must choose between a 2 or 3 way, go for the 2 way.

If there is relatively little or no difference in price between the 2 way versions and the 3 way versions...

Would it make sense to get the 3 ways instead?

Are there any advantages, in my set-up, to using 2 ways instead of the 3 ways?

In a nutshell, the difference between 1way, 2way and 3 way is audio. If your an audio aficionado, 3 way'* or even like Tom mentioned, complete components is the way to go.

As the old saying goes, "you get what you pay for".