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 Cups a bit easier, so I highly recommend it! (However, unlike Origins, Hearts do not carry over between levels.)
Many levels let you collect a Heart in the brief loading screen before the level starts. You should definitely try to take advantage of this whenever you can as it'll automatically increase your margin of error for injury and will net you five otherwise inaccessible extra lums if you can collect a heart while already having this one.
Generally, it behooves you to take out enemies personally. You will sometimes be presented with opportunities that allow you to rid of enemies via traps, but often you won't be able to retrieve as many lums that way.
There are many times where you will see one special purple lum in front of a group of regular yellow lums. Generally, collecting the purple lum will cause the next one in sequence (indicated by a faint yellow line) to turn purple, and collecting it will trigger the next and so on. Purple lums are worth twice as much as yellow ones, and you'll often need to collect many purple lums in order to fulfill the Gold Cup requirement for any given level.
If you're unhappy with the lums you've collected in any given area of a level (because you messed up some purple lum combos or what have you), you can just commit suicide to respawn at a nearby checkpoint (often at the entrance of the current area, 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.
Be familiar with the lum counts required for the highest rewards. In general, short levels with few lums require 100, short levels with many lums require 300, and longer levels require 600. Since all non-Invaded levels offer four tiers of rewards (Bronze Cup, Silver Cup, Lucky Ticket, Gold Cup), you can quickly determine the necessary counts for those rewards by using the respective increments (25, 75, and 150).
In the Wii U version of the game, Murfy can be controlled more than in some other versions of the game. If you're having trouble meeting lum requirements for some rewards, note that you can tickle many enemies, kill background critters, and put a purple highlight on bouncy pads for additional lums. This is also good for just getting extra lums towards the total lum counts necessary for unlockable characters.
The game recommends that you pull out rooted objects by their leaves, but a downward slam will pop them out of the ground faster and puts you in a good position to immediately destroy the popped out thing afterward.
Though some sections of the game may seem difficult, it's entirely possible to complete every challenge with just a single player.
Any Cages you've broken / Teensies you've acquired remain collected when you revisit a level, but everything else respawns as if it was the first time you visited.
For levels with ten total Teensies to collect, the King and Queen Teensy will always be in hidden rooms that require the completion of a short unique mini-game. Other Teensies may also be hidden throughout the level, but they won't be in special rooms like the King and Queen.
The HUD element that tracks Teensies can provide you with some vague tips on where to search for them. Most useful are the eight small Teensy heads that appear in levels with ten total Teensies. If you ever see an empty space between two filled-in small heads, that means you missed a Teensy and should search for it in between the parts of the level where you found the two enclosing lesser Teensies. The King and Queen markers are less useful due to less consistency, but in general they are often found somewhere around where you find the fourth and fifth lesser Teensies. This is by no means a hard and fast rule though, because the King and/or Queen may be in earlier or later parts of the level.
A dashing attack is great for quickly picking up speed, especially when your attack connects with an enemy or is performed on something slippery.
If you're struggling to beat the time requirement for all three Teensies in an Invaded level, you're probably missing one or more shortcuts somewhere. Be on the lookout for opportunities to shave off some time by doing something less obvious, as they may be necessary for the fastest times.
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 Invaded challenges in levels with prolonged swimming.