You can try to take a systematic approach to further pinpoint the problem with your CD player.
Take a look at the CD's that actually play in your CD player and try to determine what makes these CD's different from the other CD's that will not play. Things you can compare are:
1. Scratches, fingerprints, other foreign substances on your CD.
2. Look at the flatness of the CD's
3. Look for obvious signs of damage, such as cracks or chemical "burns" etc.
4. Make sure that the CD is secure when on the motor platform inside of the CD player. Some wear and tear could be allowing the CD to shift as it is rotating.
5. Also, on the motor platform, some manufacturers install a piece of thin rubber around the platform. I've seen this thin rubber come off on some parts of the platform, which keeps the disc from seating fully on the platform. Thus the CD is out of plane and goofy things happen at that point.
6. Check to make sure that the motor platform isn't shifting while it is rotating. Some of the screws or mounting hardware may be allowing the motor to shift.
If your CD player is continuing to skip on track 1, you may have a mechanical or electronic fault within the CD player itself. Make sure that your CD player is clean, especially the "laser" lense. If you decide to clean the "laser" lense, don't use anything but very soft cotton, such as a q-tip or cotton swab.