How should I build a system around my 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

your cousin has given you a classic set of speakers which play particularly well with moderate to high pwered ss gear. i would second the recommendation for nad, be it a receiver or integrated amp...have fun

Nice gift, the L1290's are a fine speaker, in fact they were one of the first to use a "soft dome" tweeter, a very good thing IMO.
If you are not in a hurry, go to garage and estate sales. I get all kinds of good stuff at these. I take home any working item with a name I have heard of, and at a very good price. I have found some gems at these sales. If I don't like it, after I have played with it for a while, I clean it up and sell it on Audiogon or Ebay... simple.
My buddy got a working McIntosh tube amp pre amp set up at an estate sale for $200, it's worth over $3000. Luck is a good thing to have.
The garage/estate sale track will also expose you to many brands and you may even make some money to further you to your ultimate stereo goal, what ever that may be. You may find a gem as well, you never know. There is usually very little competition for stereo stuff and albums at garage and estate sales, most people are looking for furniture, jewelry and rare books.
Hope this helps, good luck.