Current -
First, be very careful about the suggestion above regarding using a couple of AAA's - Kenom is right but he knows what he's doing and experience has given him a good feel for things like the condition of his batteries. Just be sure they are really low on juice, and don't substitute another kind of battery, or else you can fry the LD.
slightly off-topic, but I hope helpful:
I'd suggest some practice extracting non-critical items... can you get any really cheap (or free) cd or dvd drives? they can be dead ones - try a local electronics outlet, or repair shop. That way you can get better at extraction (and soldering) without worrying about killing an expensive part. Just be sure not to put power to any of those extracted diodes - or take great care when you do - because many if not most may be infrared and you can fry your eyes without even seeing the light.
And don't throw away the circuit boards from those drives - there are lots of useable resistors, capacitors, even transistors and other things on them that you may be able to use in the future (unless you have excess $$ and would rather work with all new parts).
The other suggestion starts with a question: can you get a current-regulated driver/supply to work with? If not, you're going to fry a lot of LD's. There are several people here selling various versions, some are assembled and some are kits. I'd suggest getting an assembled one until you get better at soldering. You can practice soldering on the boards from the drives you've taken apart - remove a bunch of parts, stick 'em back in, etc... modern boards may have nothing but "surface mount" parts, so if you want to practice with "through-hole" parts (the ones with wire leads) just go get some junk throw-away like an old phone answering machine or something. Of course, being a bit of a packrat, I'd also keep an eye out for useful parts in the process.
DanQ