How should I build a system around my speakers?

Given the info you provided, you just might want to consider a HK 3480. I think I saw one here on Audiogon a couple of days ago. No matter if it's here or not, it's under $500 new and should serve any purpose you might need. It has plenty of power, phono, tuner, etc and I'll bet it'll sound fine with the ads speakers.

Hi James: for economy you might go with an integrated amp vs. separates. Sorry although I have no specific component rec's: NAD has an economy product line that's actually pretty decent & they have been around for quite awhile. They also make CD players & tuners. Try to get some reasonable economy cabling like MIT T1 or T2 or T3 offerings (vs. basic plastic interconnects and lamp cord) for your source component and speaker cables. If your components have detachable AC cords then you'll benefit sonically with even a basic model upgrade AC cord vs. the stock cords.
Wish I could be more specific, but I'm kind of at the other end of the price spectrum now. Everyone has to start somewhere; I too came from very humble beginnings when I built my own equipment from scratch & from kits as a teen. Welcome to the club; you are on the right track by building the system around your speakers, which is the best approach IMO. ADS actually made some pretty decent speakers: I have ADS in my car system.

James,

Will your system be for multiple sources, like CD, Phono, tuner, DVD player, etc., or will it be for playing only a single source (like CD)?

Does your $1000 budget include source components too?

TIC

Get an integrated. for one grand you have lots of choices and can get something decent.
Larry