Icy veins has a good guide on upgrades, but the TL;DR is that adventure gear only needs Flightstones for the first half, and then Whelpling crests for the second half.
Adventure gear is rewarded from world quests and such. You can get up to Champion gear without stepping into a raid or M+, but its generally only acquired from certain group events out in the world like Fryakk assaults.
Hey there, I think this wowhead article does a good job of explaining how the system works: https://www.wowhead.com/guide/item-upgrade-system-dragonflight
As for adventure gear, it mostly upgrades with either flightstones and whelpling crests. If memory serves, I think you can get that gear from rare mobs and running heroic dungeons.
Explorer (the lowest level) and adventurer (the next lowest level) gear can come from world quests, rares, and other outdoor activities. These same activities can give you whelping and drake crest fragments, which you can then turn into the associated crests to use to upgrade your gear. You'll also need flightstones, which are earned by doing just above anything (including the activities that reward the gear).
Wowhead has an item upgrade article that goes into more detail, but my strategy boils down to world quests and rares in Zaralek Cavern, the Researchers Under Fire event once a week, the Fyrakk assault event once a week, and Soridormi's time rift events as often as I can catch them. The events in particular will award veteran (and sometimes champion) level gear, and as long as I head to the upgrade vendor whenever I'm capped on flightstones, I can get characters outfitted in upgraded veteran gear - with some champion pieces mixed in - without too much trouble.
Thank you all who posted. I'm back on track with this thing now. I greatly appreciate the help.