Here's what you need:
1. What diMeola is using.
2. A microphone, mixing console, effects processor, and speaker cabinet (speaker can be active if mixer is not, or mixer could provide power if speaker is not).
You run the FX processor as an insert, or, Aux send/Return (either to the main bus or a channel).
3. You could do the same thing as #2, but rather than a microphone, you would need to have some kind of pickup system in the guitar - either microphonic, piezo, transducer, or combination that runs out via XLR Balanced signal (usually requiring a 9v battery, but possibly running off of Phantom Power) to the mixing console.
For both #2 and #3 above, a "Guitar Effects Processor" might work OK, but some kind of rack mount FX processor would actually be more appropriate because the signal from the mixer will be Line Level rather than instrument level (which is common in guitar effects, though there are some that run at line level).
4. An alternative would be to have a similar pickup system in the guitar but that comes out as 1/4" UNbalanced signal. This could go directly into a typical guitar effects processor and on to a typical guitar amplifier.
I think you'll find though that while the processed sound would be good, any "natural" sound of the Classical will be underwhelming.
5. If you find an Acoustic Guitar Amplifier with XLR (and/or 1/4" inputs) and an Effects Loop, you could just run the guitar into the amp and put the Effects in the Effects Loop (which is usually at Line level but some can be adjusted to instrument level).
This would be better for "natural sound" than #4 above.
See, the problem is this:
Acoustic Guitars and Nylon String Guitars that work on a Piezo/Mic (maybe Transducer) kind of signal are putting out a full-range sound that can (and should) go directly to a PA system (mixing console and speakers), directly to a powered speaker cabinet for PA, or to a full-range Acoustic Guitar Amplifier designed to handle full-range signals (all the PA stuff is full-range too).
With those, you need to use Line Level (+4/-10) Effects Processors. The FX could go between the Guitar and Speaker, Amplfier, or Mixer if it's the right level. But they're usually best applied in an EFFECTS LOOP which the Acoustic Amp might have, or in the case of a Mixer, it's done with the Insert Jack (or Auxes).
Electric Guitars and their Amps run on a different level (not even really Instrument Level). If you plug an Electric Guitar directly into a Mixer or Powered Speaker, or even Acoustic Amplifier (unless it's got a dedicated input for Electric) it's going to sound horrible. Likewise, when you plug an Acoustic guitar into an Electric Amplifier, it's not going to be full-range and not going to be what you want I think. Electric Guitar Amps are not full-range and act as a "filter" basically filtering out a lot of the low and high frequencies we'd want.
That's not to say it won't work the "wrong" way, but it may not produce the sounds you want.
One final, but more cumbersome alternative would be to find some kind of pickup system to put in your guitar (soundhole mount).
Use a Microphone into a PA system on one channel, and then run the other output to the FX processor and on to the PA in another channel, essentially 2 signals from one guitar. You'd have to mute which channel you didn't want, but it would allow you to have full Classical, Full processed, or any combination of the two in infinite variations. If you had a stereo setup you could even put unprocessed on one side and processed on the other if you like. But you'd have to control the mixer (or have someone run it for you).
So everything above is have the FX "in line" which you'd just use on the guitar signal, or bypass it to let the plain guitar signal through. This last method allows you to do both.
Feedback is only an issue when the ambient sound from any speakers is getting picked back up by the microphone or any pickups.
If that's the case, it's simply too loud. You either need to move the speakers, or turn it down (you can learn how to EQ it out, but that also affects your overall tone).
So there are actually a lot of practical ways to tackle this, but nothing really "simple".
The simplest thing to do would be to buy a guitar with electronics in it and then be as satisfied with it's classical sound as you can make yourself. But it's never going to sound like a good condenser mic in front of it. Otherwise, your compromise will likely come on the "electric" side as most other solutions that are not built-in the guitar won't produce great electronic sounds - but that may be OK if the tone quality of the effect doesn't have to be "electric guitar" standards, but just "there".
If I were going to do something like this, I'd give up on getting "the best" sound out of either side, and buy one of these:
And a small mixer with Insert Jacks or Auxes, and a Powered Speaker cabinet. Then run an FX Processor of your choice in the Insert or Auxes.
If you've got plenty of money to spend, you can up the quality of each component as you see fit.
But running a microphone into an FX box and then on to an amplifier/PA just isn't going to give you very good results. It'll work - but probably not in any usable manner.