Transistors today fall into two basic categories: FETs (which itself breaks down into varieties such as N-channel vs P-channel, enhancement vs depletion mode, JFET vs MOSFET, and more) and BJTs. There are some rare, but still existing ICs which use BJTs (ECL topology) and can be used to make a CPU. But they are not commonly found. Most CPUs are based upon a complementary (N and P types) MOSFET type, commonly called CMOS as short-hand for complementary-MOSFET. MOSFETs do require momentary currents when they switch, to shift the voltage at their gates, but are otherwise field-effect based from a voltage difference applied to their gates (in reference to another terminal.)
So it's not all that useful to worry about "current amplification," per se. Besides, even if BJTs were used (and they were used back in the day) they can be applied in a variety of arrangements within circuits and aren't always used for current amplification, anyway.
BJTs are still used in many other applications, though. And there are a lot of devices that don't support much current for their signalling. Some kinds of microphones fit that, for example. Also, a microcontroller's IO pin often can't produce enough current to properly light up a higher current LED and may require some help with a BJT, as another example where a BJT's ability to accept a small current to control a larger one is useful.
To add further to your confusion, I suppose, even a BJT uses a voltage to control it's larger current capability. You'd have to look at the Ebers-Moll equation to see this fact. And a lot of people still "see" it as a small current controlling a larger one, incorrectly. But it is really the Vbe voltage on the BJT that implies a larger (Ic) collector current. The base current is just an unintended consequence due to recombination of charges. Still, that turns out to also be useful because the collector current, Ic, is exponentially related to the Vbe voltage and controlling tiny changes in that voltage would be complicated and more expensive (except perhaps in things similar to a current mirror configuration -- a fact well used in IC designs where transistors are 'cheap.') So, lemons are turned into lemonade, and the fact that the recombination current at the base is nearly linearly related to the collector current is used to good effect, at times.
You live in a modern world where you really can learn about how computers work inside without spending a lot of money. So these are exciting times for people like you with an interest. Here are some recommendations:
- Clive Maxfield and Alvin Brown wrote an excellent book for beginners here called, "Bebop BYTES Back: An Unconventional Guide to Computers." Well worth getting. And you will learn here about what kinds of basic functional units are used within a CPU and how they are put together to achieve instruction execution.
- You can also visit the NAND 2 TETRIS site at http://www.nand2tetris.org/ to get a truly wonderful approach, as well, to understanding what is going on inside of a CPU.
- For a specific design approach using FPGAs and Verilog HDL, you could pick up Monte Dalrymple's, "Microprocessor Design Using Verilog HDL." It provides a very detailed design approach for the Z80 CPU.
- Consider buying an FPGA board. I like Digilent.com as a company and, if you are in a school of any kind, you might be able to get their boards at half price, too.
- Get a software program called "Logic Friday" as a supporting tool to help you specify and then minimize complex logic functions. It's free and it doesn't do anything other than the work you'd expect from it. No ads. Etc. Just a nice, useful tool.
- Also for beginners considering the idea of learning an HDL, I'd recommend getting Douglas Smith's, "HDL Chip Design" book. It is a kind of "parallel" book on Verilog and VHDL (the two dominant HDLs) and it is targeted at beginners. There are other good books as well on these topics, such as Pong Chu's, "RTL Hardware Design Using VHDL," and K. C. Chang's, "Digital Systems Design with VHDL and Synthesis: An Integrated Approach." But you also need to learn something about floorplanning and none of these books really over that for you. (The tools do some of it automatically, but rather badly at times.)
None of the above gets into the electronics design level, though. Once you assume that there exist certain basic logic functions and you learn about the meaning of the divisions between combinatorial logic and sequential logic, the above covers a great deal. At the electronics level, you'd need to learn about signalling, different topologies and how and why they work as they do, set up and hold times, and a host of other issues that electronics designers worry about -- especially if you need to get down to FAB processing, as well. But for your level of interest and readiness, I'd probably start out with (1) and (2) above. (2) is available over the web and you can learn a lot there. (1) is a book, but it is truly designed for someone at your level, I think. So both are good places to start out. The rest is for you, if your interest continues.
In short, I think you may be getting too focused on some aspect right now that isn't important, given what you also say is your interest about learning how computers run programs. I'm suggesting you let loose of that and focus on the correct "next level down" stage and work yourself towards the rest, later. You need to take this in stages and you've jumped "way too far" into the deep end (which isn't important yet, for you) by asking about current amplification in a BJT, I think.