First, check how much storage is covered by the pictures. 800 megabyte go on a CD, 4.5 gigabyte go on a DVD, so if you have a drive with burning capacities and if they fit on a few (max 5, or 10 if you are a patient guy), you may save them there.
Online clouds have limited free space, like 2 GB or 5 GB. More comes for a fee, e.g. 7 € per month for a terabyte (1,000 GB) at Microsoft OneDrive.
The laziest option would be to extract the hard drive from the notebook and connect it to the new one with a hard drive connecting tool. This might cost
40 €(?). You can then swap any data between the drives, even use the old one as an external drive. But beware, old drives die first!