
schalten
to shift gears
To change gears in a vehicle with a manual transmission.
Du musst herunterschalten, wenn du den Berg hochfährst.
You have to downshift when you drive up the hill.
to shift / switch
Loading your saved state.

to shift gears
To change gears in a vehicle with a manual transmission.
Du musst herunterschalten, wenn du den Berg hochfährst.
You have to downshift when you drive up the hill.

to switch
To operate a switch or device in order to change a state, e.g. to change a TV channel or put a device into another mode.
Bitte schalte auf das zweite Programm, dort kommt eine Dokumentation.
Please switch to the second channel; there's a documentary on there.

to catch on
Figuratively, to think quickly, understand, and react.
Als sie die unerwartete Frage hörte, musste sie erst einmal schalten.
When she heard the unexpected question, she first had to catch on.

to rule the roost
To have full control and be able to act and decide at one's own discretion.
Als Abteilungsleiter konnte er dort nach Belieben schalten und walten.
As head of the department, he could rule the roost there as he pleased.
'Schalten' (to shift/switch) is the base verb for many important separable verbs that describe a change of state. These include 'einschalten' (to switch on), 'ausschalten' (to switch off), 'umschalten' (to switch over/change channels), and 'hoch-/runterschalten' (to shift up/down). The base word 'schalten' always refers to the core of the process.
Kannst du bitte das Licht einschalten?
Can you please switch on the light?
You can recognize the meaning of 'schalten' (to shift/switch) from the context. If it is about a vehicle, it means 'den Gang wechseln' (to change gears). With devices, it is about changing a state, e.g. a program/channel. Without an object and in the context of thinking, it means 'schnell reagieren' (to react quickly).
Beim Anfahren am Berg muss man die Kupplung langsam kommen lassen und dann schalten.
When starting on a hill, you have to let the clutch out slowly and then shift gears.
In the context of driving, 'schalten' (to shift gears) is often combined with a direction or a target. Typical constructions are 'hochschalten' (to shift up), 'runterschalten' (to shift down), or 'in den dritten Gang schalten' (to shift into third gear).
Auf der Autobahn kannst du in den höchsten Gang schalten.
On the highway, you can shift into the highest gear.
When 'schalten' is used in the sense of 'denken' (to think) or 'begreifen' (to understand), it often describes the speed of the reaction. It is therefore frequently combined with adverbs such as 'schnell' (quickly) or 'langsam' (slowly).
Er ist ein cleverer Mitarbeiter, der immer sehr schnell schaltet.
He is a clever employee who always catches on very quickly.
The idiom 'schalten und walten' is a fixed unit. It means that someone has full control and can decide freely. The two verbs are always used together.
In seiner eigenen Firma kann er schalten und walten, wie er möchte.
In his own company, he can rule the roost as he likes.
to change / switch
When referring to a vehicle's gears, wechseln and schalten can both be used. Den Gang wechseln emphasizes moving to a different gear; schalten more precisely describes the action of operating a manual transmission that causes this change.

Vor der Steigung schaltet der Fahrer in einen niedrigeren Gang.
Before the incline, the driver shifts into a lower gear.
Vor der Steigung wechselt der Fahrer in einen niedrigeren Gang.
Before the incline, the driver changes into a lower gear.