Try to always have a Heart already collected for your character whenever possible. Collecting a Heart whilst already having one (or whilst every player has one in co-op) will net you an extra 5 Lums, which allows you to increase the margin of error for squeezing levels dry of their Lums. It's skillful and it makes the challenge of nabbing the Medal a bit easier, so I highly recommend it! (Also, Hearts can carry over between levels, so you can get an extra 5 Lums in a level just for doing well in the previous one.)
Generally, it behooves you to take out enemies personally. There are a number of opportunities that allow you to rid of enemies via traps, but often you won't be able to retrieve as many Lums that way. Many enemies cough up two Lums per entity (one for the initial hit and a second for popping their bubblized form), and those small amounts can add up over the course of the level and help you reach the requirements for the Medal.
Sometimes you'll be just short when trying to reach a floating Lum, and often an upward attack will give you just the boost you need.
If you're unhappy with the Lums you've collected in any given area of a level (because you messed up a Lum King opportunity or what have you), you can just commit suicide to respawn at a nearby checkpoint (often at the entrance of the current area or nearest hidden Electoon alcove, and with a Heart if you had one before dying) and try again. There usually aren't consequences for doing so, so I highly recommend doing that instead of restarting the entire level if you make a mistake.
If you see some flora or background object emitting small bubbles, it means that it contains some hidden Lums. Jumping or light attacks can release these Lums individually, but you can release them all at once with a downward slam attack.
Though some sections of the game may seem difficult, it's entirely possible to complete every challenge with just a single player.
During Moskito levels, you can suck up an infinite amount of small enemies (like flies and baby dragons) at once. If a group of enemies are flying in a straight line, sometimes it's faster to suck them all up instead of shooting them down one by one.
During Moskito levels, you can hold down the fire button to expel a steady stream of spit bullets or tap it for rapid-fire shooting.
Any Cages you've broken / Electoons you've acquired remain collected when you revisit a level, but everything else respawns.
A dashing attack is great for quickly picking up speed, especially when your attack connects with an enemy or is performed on ice.
Although kind of jerky and hard to steer, underwater dash attacks allow you to speed through submerged areas very quickly. Though usually not necessarily in normal play, repeated dashes are useful for quickly getting through Time Trial challenges in levels with prolonged swimming.