Non-RC electric wiring questions

I like to tinker with electronics, but I do so mostly by dissecting things and making sure before I just a wire I know where it goes so that when I rewire I can make it work (aka I don't actually know what I am doing!).
I took an old fm/tape walkman apart because I wanted to dissect the AM/FM radio part of it to make a new project. I want to use that am/fm radio circuit board and attach it to an old speaker in discrete box so that I can have a radio playing without having a radio out in the open (I like to confuse people which I am sure I am already doing on this thread).
I wiring everything up and it works, but the speaker is not very loud. The speaker is rated for 40V and I am only putting 3V through the system (2-AAs).
Before I took the radio apart it was only running on 3V so I assume that the wiring and board shouldn't be pushed to a higher voltage.
So finally to my question!!!:
How can I increase the voltage (thus volume of the speaker) and not fry the circuit board?
Can I just wire more voltage in series to the speaker?
I understand basic wiring, but when you start getting lore then 5 components on a circuit board I get lost.
If you can give me the answer, or point me to a good tutorial that would be great.

Here is a question for all of you Chinese knockoff gurus:
Any idea where to find a pre-assembled am/fm radio circuit board setup for ~3-6v DC for cheap? I can disassemble an $8 radio if there isn’t anything cheaper.

Thanks for your help!

[BrianG]

I assume you mean you have a 40w speaker. Yeah, a "system" running on a couple of AA's would only be capable of ~1/2w (and that's assuming BTL on 4 ohms). 1/2w won't drive a 40w speaker well (as you've seen). What you need is a small amplifier. Just increasing voltage won't work because as you increase voltage, the current needs will increase as well.

No matter what, you're gonna need a larger power source. Two AA's is just not gonna cut it. Without some type of elaborate switching PS to boost the voltage, you'd need a 0.1ohm speaker to get 40w on 3v DC. So, you should be looking to get at least a 12-14v DC supply capable of ~2A.

Canakit sells small project-type circuit boards that might fit your needs. those might seem a little more complex than you were looking for, but the instructions are quite clear on how to assemble them. I used a small amp for my "R/C car stereo" thing a while back. In your case, 20w should be fine.

Then, since you are using a 12v supply for the amp, you can use a simple votlage regulator for the radio board. Don't worry about wasted power as the current consumption will be quite small.