Here's how to do this.
If you believe the laptop is on its last leg and that is the reason why it won't boot, then there's a possible chance that even using boot CD may not work or will just end up copying corrupt data. Reason for this is because that drive is still inside a laptop that is damaged or is dying.
Take out the hard drive. Next step would be to connect it as a usb drive to the new netbook to transfer the files. Since the netbook that you bought will NOT have a 2nd IDE or SATA port to hook up to like a desktop this means you will have to buy a IDE to USB or SATA to USB converter.
if the old hard drive that you pull out from the laptop has connectors like this

it means it's IDE and you will need to get 2.5" IDE to USB converter.
however, if it has connect like this,
then it means it's an SATA. and you will need to get a SATA to USB converter.
they are actually pretty cheap. You can order one for from $20 to $40. It will be a data cable with power adapter and power cable combo.
it will look like this.
Notice it has powercable, and adapter, and has a converter with usb cable on one end, and sata cable on the other, and this device also has 2 additional ports for 3.5" IDE (desktop hard drive) and 2.5" IDE (laptop hard drive) and it will come with instructions. You do NOT need to install a driver for it since for many years now USB drives have been plug n play. Just connect your drive, plug the power, and then connect the usb portion to the new computer. this device is listed as being $22.84 at
PC Connection - SATA IDE to USB 2.0 Adapter-Cable
I do however recommend spending just about $15 more and getting a real nice docking station for SATA if it turns out your old laptop hard drive is in fact an SATA. I have one that works with both Desktop hard drives and laptop hard drives and it's great. this is what I have or similar to what I have. this item is listed as $39 at
PC Connection - BLACX 2.5 3.5" SATA Hard Drive ESATA and USB Docking Station
and looks like
Now on the note of preventive measures. THis advice is for everyone.
There's nothing like having a backup, backup, backup. I highly recommend having either a device like above with a nice size 1TB or 1.5 TB Seagate hard drive which costs about $180 or less these days. Or you can buy a WD passport drive if you need portability.
With first hardware solution you get a pretty good size storage for your money but you will spend a few bucks on that docking station. If you go with getting a very light weight portable WD passport usb drive, you will not get as much storage space for the same bucks cause it's hard for manufacturers to fit so much storage into a tiny hard drive like they can in larger desktop hard drives.
Now for software. I will recommend 3 different backup softwares that do different things.
1. HandyBackup - probably ideal solution for beginners and anyone else who just wants a simple file only backup solution. This backs up all your documents, pictures, music, and anything else you want backed up on schedule bases to destination you specify.
2. BeyondCompare - this software is a bit technical but it's so full of features it's worth the money. It allows you to sync 2 different folders (for example, your C:\users\JohnDoe folder to your Usbdrive\laptopbackup) and so if you do this regularly you will have exact duplicate of all your files AND the folder structure of your laptop data in an external usb drive.
3. Acronis Backup & Recovery 10 - I currently own this but it costs money. I paid over $100. Yes I love it that much. It backs up your hard drive partitions. It does full, incremental, differential. It backs up not just your documents but it is mainly for backing up your whole hard drive including your operating system and ALL its installed software with all the registration activated and such. Reason why this software is so good is imagine that your hard drive out right burned up and crashed. You can spend hours installing os, and ALL the software AND re-activating and THEN restoring the files
OR
You can just use Acronis Backup & Recovery 10 and take as little as 8 to 10min restoring your whole hard drive data to a new drive in the same exact structurer you had the old one. And Just reboot and you will be back up and running with your windows working and activated. all your software installed and activated, and ALL the settings that you changed still being set the way they are supposed to be and most importantly all your documents.
Yeah I do this for a living. Have a nice day.