Replacing rear solid axle with skewer Hoping someone can advise. 

I recently ordered a Saris Fluid2 trainer, which requires me to use a special quick release skewer on the rear wheel.  However, my regular road bike is an old Acapulco Giant and the rear wheel is not on quick release hardware at all (photos below). I am not much of a cyclist and even less of a mechanic, under normal circumstances I'd just bring the bike in to my regular shop and have them do the switch, but at the moment we're completely isolated here at home.  Would really appreciate whatever advice you can give on doing the switch. Do I need to worry about the gears coming apart when I disassemble? Do I need any adapter hardware? What else do I need to know?

Thanks in advance

edited to add: I see that someone has already asked about using a trainer with an existing solid axle.  That is a super helpful question and answer but my original question - now also very well answered - was about swapping out the existing axle for a different one.
Leave the wheel in tact and see if you can order some 'turbo trainer axle nuts', pictured below, to replace your current nuts on existing axle. If these don't work for you then start disassembling the wheel but research first; you need skinny spanners called cone spanners (£?) to do the job properly and risk dropping ball bearings (annoying). Not normally a huge problem but if bike shops are closed and we're all social distancing atm, then you want to reduce the risk of messing up, temporarily, in case you are left without a functioning bike or help to fix.

Trying to patiently find a solution with low risk atm is reasonable.