I've been using three-part cycling glasses for a couple of years.
That's a frame, separate plastic insert in front for colour, and a small inset frame in behind for prescription lenses.
What's the best way to care for them and maintain them, both while riding and after the ride ?
The three parts unclip from each other easily enough. I've never broken the front part, though the lens-holding frame only lasts 6-12 months before cracking and splitting.
Oakley has always supplied microfiber bags with its glasses, and it recommends wiping them using only those bags. Their care instructions are linked here. I believe that many other premium glasses manufacturers also supply microfiber bags. Rougher cloth will scratch the lenses. Many premium sunglasses have relatively delicate coatings that are best handled with microfiber cloth. The bags should be washed by hand or on the delicate cycle - and because they are picking up dirt and oils, they do need this occasionally. In addition, I would recommend storing the glasses in the bag, since even incidental bumps may scratch the lenses - these won't be safety critical, but many of us prefer not to have our fancy lenses scratched where possible. A hard case may be useful, but probably not necessary.
For a more thorough cleaning, I believe washing with dish or hand soap is sufficient and accepted by the manufacturers. This applies to both the lenses and the rubber ear stems and nose pieces. I've seen cleaning sprays offered by the sunglasses manufacturers, but I don't currently believe they're worth the premium over just soap and water. One thing to note is that if you can remove the ear stems and nose pieces, you should occasionally do so to wash underneath (e.g. yearly). Over time, build up from dirt and sunscreen accumulates under the ear pieces.
It can be tricky to get the ear stems off and back on. For Oakley glasses, SportRx (an online sunglasses store) has a YouTube video showing general principles. Other manufacturers are likely to follow similar principles.
I'm not certain what recommendations to make for care during the ride. If possible, I'd avoid wiping the lenses on my clothes if the lenses get soiled for reasons stated in the first paragraph. Naturally, if it becomes safety critical, disregard this. I'm aware that some manufacturers have oleophobic (i.e. repels oils) and/or hydrophobic coatings (repels water), but not all do, and I don't know how well they work. Oakley, despite their premium price, doesn't offer these, so it's possible there's some sort of trade off with performance. You may be able to buy coatings to apply to your lenses. This might be worth considering if you frequently ride in precipitation. However, I've never used them and can't offer any first hand experience.
Depending on your manufacturer, it may be possible to replace a damaged lens, or to get a third-party replacement lens (e.g. from a place like SportRx). The third-party lenses may not fit exactly due to manufacturing tolerances; I had this issue once, but the fit was acceptable. If you had a premium lens (e.g. Oakley's Prizm lenses), the third-party ones may not quite offer the same benefits (e.g. they may skip high end coatings). This may not be possible for all glasses. Removal of the old lenses or insertion of new ones often requires what seems like a lot of force. There are probably YouTube videos for your model of sunglasses if you are unsure.
It is often possible to get replacement ear stems and nose pieces as well. Rubber does lose its elasticity over time, so long-term users (e.g. over 3-4 years) may need to do this at some point. I'm not sure if all manufacturers stock these spares. To my knowledge, Oakley makes the parts available for most or all its models (although they may discontinue models). Tifosi, a mid-range cycling sunglasses brand, also lists spares on its site. For budget brands, it may vary, so check with your manufacturer.
Eclectic advice:
Now for the "method": rinse with cold running water, and follow with exactly one 1. firm, 2. from end-to-end, 3. with a perfectly clean cloth, swipe. That's all you need. Inspect against light. If you see streaks, repeat at most one more time, or else you really need to wash the lenses with soap first.
The regimen above is just for the lenses. The lenses themselves rarely need to be washed, but oil often accumulates on the temples from the sweat of the wearer. You can wash and dry temples as much and as often as you want, just so long as you isolate them during washing and that you apply a more gentle regimen for the lenses.
This is all for the before/after care. The best you can do during a ride is to wick sweat by touching the edges of the lenses with your jersey.
In addition to Weiwen Ng's answer, for quick washes when I don't need fully-clean lenses I'll just rinse my cycling glasses under a faucet and then dry them off carefully with a paper towel. Most times, that's good enough because it won't be long before they have sweat drops all over 'em anyway.
That admittedly might not work too well for the glasses pictured in the question, with the two layers of lenses.
I'll also specifically recommend not using cleaners like Windex - the chemicals in cleaners like that can and will eat away at any coating on the lenses.
I find there are four main sources of stuff on the lenses.
While riding one can temporarily remove rainwater by wiping the front with an index finger, from nose to outside, one lens at a time. Rainwater is generally clean so low risk of scratching. Reduce incoming rain using a cycling cap/casket aka a visor/brim on your helmet.
For sweat, it is salty and will leave deposits behind if you wipe. A sharp "nod" motion helps sweat to drop off the bottom lip, and any sweat/salt is mostly out of the line of sight. You can also help divert sweat with a headband or similar.
Fog forms for me when riding hard in the cold and then stopping at lights. Wiping doesn't help here cos the fog is on the inside surface, and trying to wipe results in awkwardness and the risk of dropping them. Instead, pull the frames down the nose a bit to let the cold air in easier which helps equalise the temperatures. As soon as you start to move, the cold air clears the fog though this can be a bad time to have impaired vision.
Dirt risks being abrasive - while riding its best to stop and splash some drinking water on the lens. Don't wipe dirt/mud off with a glove.
After riding I've been told that any chemicals/cleaners are bad specifically for the frame that holds the lenses. As such, I wash them off under a tap or in the shower. The three pieces are separated for easier access.
Shower-temperature water has not caused me any problems in the past, but the inner frames are not very durable anyway and show cracks within 6 months. I have bought plenty of spares and figured out a safe way to insert the lenses (involves one's mouth, teeth and lips have a fine degree of sensitivity for the pressure one can exert !)
Dry with a clean cloth or paper towel in a patting motion, not a wiping action.
Edit - just after writing this, my inner frame let-go again.
This happens every 6-24 months depending on how much riding I do, but is unavoidable. A crack is visible through the clear plastic, and after some months it breaks completely.
I've managed to move the lenses to a replacement inner frame using a combination of firm-gentle pressure and luck, and I buy the inner frames from Aliexpress for a few dollars.
Rest of frame, for context.
The prescription lenses cost twenty-times what the frames did, so they're the part that needs protection.
