My advice would be to forget about LM386 and try to use a more modern audio amplifier chip.
Why? Well, it's an easy to use IC but it's after all a class AB amplifier which means it's kind of inefficient ( with 10v in and 0.2A you basicaly give it 2 watts of power so the output is probably going to be less than 1.2-1.5 watts in a 8w speaker, not very loud) and a lot of power will be wasted.
Also, there seems to be two "versions" of lm386, one that works at 4-12v and one that works at 5-18v - if you use the first one, the people you give this to may use 8 x 1.5v alkaline batteries (which have about 1.65v when fresh) and kill your device.
I assume you chose LM386 because it's available in DIP package (the one with long pins, easy to solder on board) and probably because there's a schematic available already on lots of sites. If you want DIP package, I'd suggest looking at a chip like TPA3122d2n :
http://uk.farnell.com/texas-instruments/tpa3122d2n/ic-amp-audio-pwr-15w-d-ster-20dip/dp/1755372It's a class D stereo chip (but can also be configured as mono) and there's schematics inside the datasheet for both modes of operation.
It may look scary at first but if you read the datasheet, it explains there how to choose every component and what's important.
There's also a tutorial made by Afrotech on Youtube where it uses this exact chip :
http://hackaday.com/2014/06/09/afrotechs-guide-to-class-d-amplifiers/#more-124134 (the video is in the page)
This IC needs at least 10v to run (and it would prefer more). As for how to power it, I would suggest using a lipo battery from sites like Hobbyking :
http://hobbyking.com/hobbyking/store/__86__85__Batteries_Accessories-Li_Poly_All_brands_.html They may seem more expensive compared to 8 rechargeable batteries but they hold more charge and you could also use a very basic lipo charging chip to provide charging capability to your project. There's not many in DIP packages, but there's a lot of that that can still be easily soldered onto a pcb and don't need lots of parts, for example something like this MCP73812 :
http://uk.farnell.com/microchip/mcp73812t-420i-ot/ic-li-ion-li-poly-charger-sot23/dp/1627187 Basically, you plug a usb cable at the chip's input (it can handle up to 6v) and the chip charges the lipo battery at up to 450mA and stops when the battery is charged.
Of course, the battery would only give you about 4.2-4.7v so you'd need something to boost this to above 10v. There's plenty of switching regulators capable of working from 4v to boost to about 12v which decent efficiency.