I'd appreciate if you tell me what is the right way to say my bike/motorbike/car's tire is running out of air and needs to be filled with air again in English.
We say the tire pressure is low. Also, if the tire is punctured, we say it is a "flat" or is "deflated."
To get the air into the tire we say that we need to put air into it, pump air into it, inflate or fill it.
In the UK, it's common to say a tyre (note the spelling!) is going flat.
If it needs to be refilled, a common expression might be "It's going flat and I need to get it pumped up" or "...and I need to pump it up".
We don't normally mention "air" explicitly, because it's a given that tyres are supposed to be full of air under pressure. And we use pump because — despite the ubiquity of garage air lines — the mental image is of one of these: