I've said it many times and will again. First and foremost, tone comes from your hands. So spend time playing. Tape yourself. Find the mistakes and work at it.
Listen to a guy like David Gilmour. IMMACULATE tone whether playing through his touring rig or just an acoustic. A lot is in your hands - pedals alone will not cover bad playing or make your tone. They HELP but not alone.
That said, a decent guitar and amp. A guitar that is setup well - no buzzing frets - good action. An action not too high or low that makes it difficult to play. Again, a lot of tone is from your hands. IF you are struggling, it will show (and sound).
Another often forgotten component - Quality cables. Instrument cables, pedal to pedal cables, effects to amp cables... etc. That doesn't mean you have to spend tons on Monster cable. But the vat of cheap no-name cable near the register at your local guitar shop probably isn't the answer.
Now that you are paying attention to you -- your playing -- working on paying clean and getting good tone from the playing AND you have a decent guitar AND you have decent cables AND a decent amp... For Edge-tone....A good modulation delay is essential.
The EH Deluxe Memory Man modulation had chorus/vibrato. The Vibrato section is the one Edge used.
Per The Edge, the modulation is what took the tone from 2-dimensional to 3-dimensional.
TC Electronics Flashback Delay is pretty nifty. AND it has 'Tone Print', where you can download settings. Nice. Can mock a 2290 AND does dotted 8th -- very useful for Edge-tones.
$169.00
TC Electronic Flashback Delay and Looper Pedal at zZounds
TC Electronic Flashback Delay - YouTube
Eventide Time Factor is pretty awesome. Pricey but nice.
$399.00
Eventide TimeFactor Delay Pedal at zZounds
It even got Guitar World's "Platinum Award" in their review.
Eventide Time Factor - YouTube
Strymon Timeline is also pretty great. And has built in shimmer (octo) too.
strymon » TimeLine – Multidimensional Delay Effects – Delay Pedal
Strymon TimeLine - Multidimensional Delay Effects Pedal - YouTube
As for OD, Boss SD-1 or a Ibanez TS-9 Tubescreamer are good starts for main overdrive.