Weighing in too, which I should have done way sooner haha apologies on the delay!
I'll first write up what I originally had in mind (this doesn't have to be the concrete way to go, just was the plan before this discussion):
Relevant QnA Stream snippet.
3 systems here. 1st is going to be Achievements, which to make sure we stay 100% clear on terminology, is just Steam achievements. These are fun things to try to do, and are effectively meaningless (as people can just use some program to unlock). But while fun for completionists, it ideally has an introduction to some milestones (jump 1, 100, 100k, 1m etc times) that would otherwise be meaningless in a competitive setting (who really cares if you jump 1m times, you just play Momentum enough for it to happen). They won't be server-verified, and just be fun things that casual players can try to get and brag to friends about.
2nd is going to be milestones, but a common internal term used is "Challenges" as well, and I'm not yet settled on what the term will be, but "achievement" is more syntactically synonymous to "milestone" than "challenge" (you don't "achieve" a challenge), so I'm leaning towards calling them Challenges. These are one-time unlocks (important for selection of what is a challenge), directly related to specifically crafted goals around game modes, to promote and track progress in each one. They will be giving XP and for some, specific cosmetics, on completion. Think MW2 challenges, if you've played that. Things like beating specific amounts of tiered maps, getting specific "medal" times on maps, etc. The difference here being that Achievements were pretty moot and specific, challenges can be tackled typically from different angles and are usually over time than just a one-off thing. I don't think challenges should tie in with specific maps, because we'll have...
3rd is where Achievements and Milestones fail to capture: temporal (daily/weekly/monthly) tasks. What can be something fun to bring people back every day or week or month? These are the ones that can absolutely be map specific goals/tasks. Something that lets players explore the game in the appropriate sized chunks they want to, and can get a fun, somewhat-exclusive reward for it. In the QnA stream it was suggested like "beat X trick on X tricksurf map", so think general "tasks" that can range in difficulty but provide enough of a motivation through the rewards to warrant consistant playtime over a time period. They can sync up with maps that aren't as popular to promote more playtime on seemingly underrated content, and/or they can be themed for specific holidays or seasons. They'd be too small and temporary to be considered a challenge or achievement.
Now, reading what you guys have suggested.
TheNerdman - This is pretty much what we're thinking, though Achievements will not (and cannot unfortunately) be server-side unlocked. I mean you probably can unlock them via Steam's WebAPI but if people can also do it locally there's really no point.
FryingDutch - I think you're mixing up general stats that we want to collect with milestones as well. People will feel really good about being top X in a game mode, that in itself is an accomplishment. We can maybe give a badge, but there's no way to make an achievement or milestone fit, as the moment they are no longer top X or have completed X% of maps (new maps get added/new people start beating their times) then the reward gets yanked from them. We can have a timeline of PBs and WRs but it's also just more stats, not really a rewardable thing -- the accomplishment here should be the award really.
rio - It seems like rio is on the same page here. We absolutely should do a fixed number of things to warrant any unlockable content. Having done something is something the website will track and can reward, having something temporary like a rank or % completion will not last and be "revoked" so to speak, which really sours/begs the question of why give the reward. But someone should be able to play just one game mode, get all the challenges, achievements, etc for it, but then see the rest of them and be like "huh I guess I'll try map/gamemode X", repeating the cycle for another mode in the game.
As for what specifically, that's yet to be determined. Achievements we have a nice head start, and I'm sure we're going to add like another 30 or so for the campaign, but challenges and dailies are things we'd love input for. I'd also love your input on what I wrote, sorry about the wall of text!