Cleaning takes under a minute if you do it straight away, and becomes a chore if you leave it. Here is the whole routine.
After every use
Rinse the inner liner under warm water as soon as you have finished drinking. Sugary drinks dry into a sticky film that is far harder to shift later. Most of the time a rinse is genuinely all it needs.
When it needs more than a rinse
Warm water and a little dish soap with a soft sponge. Work gently around the inside. Avoid anything abrasive, since scratches in the inner surface collect residue and are impossible to undo.
The one important rule: no dishwasher for the liner
The inner freezable liner should be hand washed only. Dishwashers run hot, and sustained heat degrades the cooling gel sealed inside the wall over time. A cup that stops holding its cold has usually been through a dishwasher. The outer shell is more forgiving, but hand washing the whole thing is the safe habit.
For the same reason, do not microwave it, do not soak it in boiling water, and do not leave it in direct sun in a hot car.
Drying and storing
Let it dry fully before it goes back in the freezer. Water left inside will freeze into a layer of ice that has to melt before your next drink can touch the cold wall.
Dealing with lingering colour or smell
Strongly coloured drinks can leave a faint tint. Warm water with a spoonful of baking soda, left to sit for fifteen minutes and then rinsed, usually lifts it. Avoid bleach and avoid very hot water.
The gel never touches your drink
Worth saying plainly: the cooling gel is fully sealed inside the wall. It is not something you clean, and it cannot leak into what you are drinking. You are only ever washing the drinking surface.