I wouldn't open any com chests till you're character 25+. The color of the items you get from chest is based on your character's level opening them. So a lvl1 character will get grey items only. But really it doesn't matter too much, and I'll explain below why.
The recruit chests are all 100% garbage. Same for the vet chests. So open all of them right away.
Basically the type of chest determines the max power item you can get from a chest. A recruit chest has 100 max power. The absolute max power you can get from a chest is 300 at champ and legendary.
Now for your item's power it's determined by the highest level item you've seen. So if you get an item that's power level 90, you can get an item that's power level 100 next time for example. This is referred to as "has seen". Your has seen is shared by all your characters, so when you start another you can start pulling and crafting items just as strong, they'll just be grey items because the character level is low. That's why it's good to save most of your chests till 25 to open on a character. At the same time opening chests doesn't hurt you because you're advancing your has seen level, it's just not optimal till you're character is 25+ for loot quality reasons. It's a balance between advancing your has seen level and getting the best loot. Personally, I think it's fair to do something in the middle to get higher has seen faster. You're probably not going to get many reds anyway!
As for advancing through the difficulty it really depends on the player. As a rule, you want to play veteran when your item level is 100, then champion when your item level is 200 because if you don't you're wasting time not advancing your has seen, but that doesn't mean you can't play the higher difficulty sooner. Going off the in game recommended hero power works fine. I would get out of recruit immediately and veteran just as soon as you can. Champion your hero power should be as high above 300 as you can. Legendary hero power 500+, but 600 (max) is definitely preferred.
I wouldn't play champion till you're at has seen 200 though, just because you're be frustrating yourself in champion and still advancing at the same rate. Higher level chests advance your has see at the same rate so it's a waste to open them till you've progressed past it.
When you do get to champion, save all your Emperor chests till your character is 30 and your has seen is 300. This can take a while but it's the best way to get red items.