Destiny 2 – How to Level Up Guns to Match Power Level

I am going to answer this from personal experience from leveling up 6 characters, it is possible there is something I'm overlooking or that theoretically there is a faster way if you maximize time.

The fastest way I've found is to first finish the story. Each story mission gives you a lot of experience. You don't even have to kill all enemies unless it specifically tells you to, you can avoid them and just get to the end. But this will mean you might have to do some adventures or public events to level up enough to access the next story mission.

Once you beat the story and get to level 20 your next mission is to get to raising your power level. "Powerful gear" AKA Luminous engrams are your main goal, these will give you legendary (and sometimes exotic) gear at a power level higher than your current max. You can get a number of them each week by completing milestones. An easy one to get is by joining an active clan and once members of the clan complete weekly milestones (even without you being there or online) they will unlock and give you the rewards. There's also exotic quests which lead you to guaranteed higher level gear and unique exotics that you can only get through the quests.

After climbing high enough you'll be able to play end game activities such as Nightfalls or Raids. Completing these weekly will not only give you more luminous engrams but these kinds of actives also have a higher chance at giving you exotics. Rise and repeat each week getting as many luminous engrams as possible and with enough time (and luck) you'll be at max power level soon.

No, cooldowns are only displayed with a white line slowly filling up the box until it is filled and changed to a colored box correlating to your subclass. There is no way to change it to a numbered countdown like those of MMO's or MOBA's.