Electronic – Homemade Electromagnet from Microwave oven transformer

There is a little amplifier I am very fond of which I have used with dynamic mics. Since your coil situation is much like the one faced by someone trying to amplify a dynamic mic, I encourage you to try this circuit and see if produces a usable signal. Just replace the "microphone" (speaker) with your coil.

http://electrosuite.com/audio/bc547-dynamic-microphone-amplifier.html

The lower transistor acts as a common base amplifier, which is fairly unusual. It has a current gain of less than 1, but it does provide significant voltage gain. The upper transistor is in a common collector configuration, so it has essentially unity voltage gain, but the current gain is good. In short, this first amplifies the voltage, then the current, to give you a strong output signal. Although you will likely need a second stage after this, that should be much easier.

EDIT:

Concerning your noise issue, do you know where the noise is coming from? Sometimes I just have a power supply that's super noisy and ruining everything. I like low budget solutions, because of, well, my budget, so I usually test this by running the circuit on batteries. If things start magically working, then it's time to work on the power supply.

Both windings are just coils of wire -- giving them different names does not change how they function.

Yes, when wound on the same core, they will definitely interact with each other quite strongly. However, you need to decide whether that interaction works in your particular application.

I take it you want to put a DC current through one of the coils in order to provide a field similar to what a permanent magnet produces. That's fine -- the other coil will not pick up DC. However, any AC (noise) in that DC current will be coupled to the other coil, and it's possible that it will overwhelm the signal from the guitar strings that you're looking for.

So, how good are you at building low-noise DC current sources?