What's the best way to clean the CD/DVD player & disc?

You get a microfiber cloth to clean the disk, and a lens disk like this. The lens disk has little bristles that sweep dirt off the player.
You can also (not officially recommended) simply wash the disk with hot water and soap, dry with a tissue (non-lotion/aloe/whatever) and then blow the tissue particles off.

In reality, 25 years of using CD players, have never once found that the lens of the laser needed cleaning. (FWIW I am a filmmaker and former broadcast engineer)

On the other hand, the DISKS do need cleaning regularly. To clean a CD or DVD:

1) Use a soft cotton or microfibre cloth. DO NOT EVER use a paper towel or Kleenex tissue or toilet paper. You CAN use “lens cleaning cloths” as you would for eyeglasses or camera lenses.

2) Use a gentle cleaning agent. Windex is fine, though do be careful as some cleaning agents will damage the printing on the printed side of the disk, especially small run inkjet printed disks. For this reason, I only wipe the PRINTED side with a DRY cloth.

3) On the NON printed side (which is the side the laser reads the data through), clean by wiping the MOIST cloth from the CENTER (hole) to the OUTSIDE EDGE.

DO NOT WIPE IN A CIRCULAR MOTION.

Wipe only from center to edge, and rotate the disk a little after each wipe until you have cleaned the entire disk.

NOTE ONE: The data is actually contained in a lacquer layer on the PRINTED SIDE of the disk, but is read by the laser through the CLEAR SIDE of the disk. As a result, damage to the PRINTED SIDE can make the disk UNREADABLE, if that damage goes through the printing and into the vacuum metalized (aluminum) coating.

NOTE TWO: Blu-Ray disks are much MUCH more sensitive to small scratches, and should be handled with greater care. I recommend that Blu-Ray disks ONLY be cleaned using “automatic” cleaning devices. If you DO clean the Blu-Ray disk manually, blow it off with compressed air first to help prevent dust causing scratches as you clean.