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.

'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
'wechseln' and 'schalten' can both mean that you go from one state to another. 'wechseln' is more general: you can switch the channel, the topic, the seat, or the gear. 'schalten' is used more when operating a device or shifting gears in a car. That is why you may hear both words with TVs or cars, but 'schalten' emphasizes the operation, while 'wechseln' emphasizes the transition to something else.

Vor der Kurve schaltet Lena in den zweiten Gang.
Before the bend, Lena shifts into second gear.
Nach der Werbung wechseln wir den Kanal.
After the commercials, we switch channels.