I cannot print or scan with the hp1315. Upon start-up I always have a test page printed. Any solution to these?

The testpage is by design, the unit is meant to be always on, because the fax needs to be always on, right? When not in use the unit goes to standby mode with very small power consumption. Consequentially, if you turned it off, upon turning on it is presumed that it was not used for a longer time and it has to be made certain, that the print heads are OK to print the incoming faxes (otherwise you wold get blank or distorted fax printouts), hence the test page.

Now regarding the printing and scanning, did the unit stop printing/scanning? Or it never did print/scan on this particular computer yet? Has the system been reinstalled for example?

Generally, you have to have appropriate drivers and software installed from the accompanying CD to be able to use the printer. If you lost the CD, you can download essential drivers and software on www.hp.com support section. Please remember that it's always best to first start the driver installation and then plug in the USB cord when the installer tells you to.
If the printer worked before, try a different USB socket and a different cable.

The "test" page is in fact a printhead calibration procedure. You have to put it on the scanner and press the scan button. This is what is shown on the page, if you read it. After successful calibration you will be able to print.