Agreed with Francois, the easiest and safest method would be to remove the hard drive and put it in an external casing, then just mount it to another Mac as an external drive and start copying the data.
If you want to use all the apps and data as-is from the old machine, you should be able to tell the new Mac to copy those from the old one and put them in all the right places. In the old days we did this with a Firewire cable to the old Mac, but if your power is sketchy you'll want to try the external drive method.