It's not actually a serious problem but to move forward you will need something more solid. A lot of snakeoil is sold in the form of supports. Very many of them make no difference to the use of a solid table.
Put the CD player underneath the amp rather than vice versa; this is all that the manufacturer is really suggesting. It's partly for safety reasons - Amps generate more heat than CD players.
In the long run, to develop the system further, look for something solid with separate shelves and no castors. Make it yourself if you have to.
People usually start to buy serious supports on the second full upgrade and the good ones really work. The ones that I think are audibly effective are anything by Townshend, Nordost Pulsar points and anything by Mana. All unjustifiable for entry level but the results can be surprising from mid-fi upwards.
One thing that you might like to try is the Ikea 'Lack' coffee table. It's much used by Linn users for their turntables & you could try it with your CD player for the cost of a moderate lunch (£8 for the side table).
Leaving them on standby is sufficient. An half hour warm up will get you to the same place. Yes, it's generally a good idea unless the piece of equipment you're dealing with is Class A or valve (none or yours is though). Try to see if you can tell the difference between when it's warmed and when it's not - if not leave it off when not using.