Desynching is being able to control both Ice Climbers at the same time, by using different moves. You can keep a desynch going as along as the moves' lag overlaps. The simplest ways to desynch are the spotdodge and roll (and grab) desynchs. Basically, you do spotdodge/roll, then input and hold down the move that you want Nana to do.
Spotdodge is really only useful as an approach desynch because the Ice Climbers' spotdodge leaves you pretty vulnerable while there are two of them. Roll desynch can be used closer because it gives you extra space from your opponent. However, it still leaves you somewhat vulnerable so I wouldn't use it too much.
More advanced desynchs include Pivot desynchs which are faster than the others, but don't give you any invincibility frames, and are harder to do. You can pivot any move with Popo other than Blizzard (for any A moves, Nana will dash attack). You can also pivot Popo-Smashes using the c-stick, and Nana will be free to do anything.
Desynchs can be used as approaches, in combos, and to help you land grabs. Desynch combos just take a while to get used to, but one of the easier ways to do them is to desynch from an aerial (not too consistent). I tend to pivot during aerial chains because that can cause Nana to hop by herself either making Popo smash or letting him follow up the aerial. For more on desynching, read exarch's guide.
I don't know what else you're looking for, so I'll just list some stuff.
Good approaches: Desynched Blizzard, Ice Block Chaser, wavedashing/dash dancing until you see an opening for a grab or smash.
Combo starters: GRABS, u-smash, u-tilt, u-air, n-air, jabs, and sometimes d-smashes or b-airs.
Combo finishers: GRABS, b-air, any smashes, u-air.
Things to help you land grabs: Jab-cancel/shieldgrab, wavegrabbing, Ice Block Chaser, solo Nana-Blizzards, n-airs, plain ol' shieldgrabbing.
Grab combos: Err, watch Avalanche to learn chain grabs. You've also got the infinite, and you can do any smash out of a throw with Nana. You have access to any combo starter if you can't chain throw them.
With solo Popo, almost always d-throw. You can smash out of it, or just chain throw with it. U-throw is good against spacies at low percents, and f-throw/b-throw can be okay if you're near the edge.
Also, perfect wavedashing! Ice Climbers have the second best wavedash in the game, but have awful traction. Therefore, you need to wavedash and waveland fluently to use them to their fullest, and also need to practice wavedash to running transitions to make up for their poor traction.
Other than that, to be good with the Ice Climbers it just takes the same skill it does for other characters (SHFFLing, mindgames). Good luck.
Edit - For recovery, try to always use Squall Hammer if you can, and airdodging if you're close (tap B really fast to go high with Squall). Belay is laggy and you can't sweetspot with it, so it's not the best recovery move. Only use it if you're too low for Squall Hammer, or to cancel a meteor.
Edgeguarding is interesting with the Ice Climbers. You've got a lot of options open to you. Just experiment until you find your ideal combination. I personally edgehog to ledgehop/b-air most frequently, but there are other good ones. For example, desynch -> Nana-Blizzard/Popo Smash is good if they miss the sweetspot, and they can't ledgetech if you do this. Simply charging a d-smash can be effective. A fun one is facing away from the stage, up+Bing out of shield and holding backwards so Nana grabs the ledge.
Just practice and try out all of their moves until you find a playstyle that fits you well.
