I mean, why wouldn't my brand new 1018 do the job? It sure seems like it can. Earlier someone suggested I buy the 663 with an amp, and I didn't give it a second thought but now I've heard it several times. I'm confused.

Just what will an amp do for or me/my new system?
It will come down to which speakers you buy and how loud you like to listen to your system. Speakers present a certain electrical load to an amplifier (be it the ones in your receiver or external amps). The louder you listen, the more current the amp has to supply to produce that level of sound. If an amplifier can handle the load, then you won't notice any distortion or clipping (i.e. everything will sound fine). If an amp can't handle the load, then you will may distortion and clipping, and things might sound less dynamic than they should.
For a real world example, take my system. I have a Pioneer 1015 and NHT speakers that take a bit of juice to play loud. At my normal listening levels, the Pioneer works great. However, at levels louder than I normally listen, the sound seemed less dynamic and muffled - the Pioneer just couldn't handle it. So, I bought a separate amp, and voila - the dynamics were there at loud volumes. So, to sum up...for about 99% of my listening, the Pioneer was perfectly adequate. I really only got an amp because I was curious, and it did help out when I crank it.
As a bit more detail, adding a two-channel (stereo) amp like Dave mentioned also helps for movies. Because it removes part of the load from your Pioneer, the Pioneer is then able to handle the center and rear speakers with more ease.
If you get speakers that are easy to drive (high sensitivity, not very low impedance), then that 1018 will probably keep you happy all day long. So, don't worry about it. Getting an amp is a bridge that we can cross when/if you ever feel the need for one. Get your speakers first.