The Controls, Cable and Amp
An electric guitar isn't a complete musical instrument -- its sound is the result of the player's skill plus all the components that comprise what engineers call a "signal chain." The chain begins with the pickups and the internal wiring of the guitar, principally the controls which are technically called potentiometers or "pots."
The control circuit comprises what's called a "load" on the pickup. That load has a tremendous influence on pickup performance, and if you connect one of your pickups directly to the guitar's output jack and then play as usual, you'll hear instantly that! Pickups work differently with different loads, and we can alter the load with different pot values or by adding shunt resistors on a per-pickup basis. Both of these modifications are far less costly than changing pickups and can be extremely effective.
The next link in the chain is the cable with excess capacitance -- a category that includes some of the fanciest and priciest brands on the market, by the way. This cable can rob your sound of much of its beauty and clarity. A good low capacitance cable is often more of an improvement than a whole set of new pickups.
Finally, there's the amplifier with its array of control knobs and its loudspeaker. It often astounds me when people tell me what amp control settings they use. I can sometimes save them hundreds of dollars by advising to simply readjust the knobs at more sensible numbers because settings vary widely from amp to amp, even within the same amplifier brand. The only general advice I can offer here is to try lots of different settings, and you might get the tone you're aiming for without spending one dime more.