If you're not technically inclined you can take it to a local PC Shop (Staples and Best Buy offer data transfer services). If you are knowledgeable enough you could remove the old hard drive from the computer and hook it up in the new computer. If this is impossible (no IDE ports or it's a laptop) then you can always purchase an external case that allows you to put the drive in and hook it up to the new PC using USB. Make sure to look for an external case that has IDE as the internal interface
like this or
this.
Another solution is through network transfers... I haven't done them myself (most of my computers I've had to do this for where Windows XP) so I'm not sure how rough it would be to set up File Sharing to a Windows 7 or Windows 8 PC. Most of the time you will have to set up another user account on the Windows 95 PC with the same Username and Password as the Windows 7/8 PC.
Keep in mind that pretty much all programs installed on the Windows 95 machine will probably not work on the new computer.
I personally prefer the External Drive IDE -> USB adapter transfer method. It's as simple as just removing the old drive, putting it in the external, and hooking it up to the new PC/Laptop and transfering what you want.