Can you link an Andoird phone into your home internet without having to pay for a monthly service? I have never had a phone before. Can phones do this?

I want an Android phone.

I want it to be able to download stuff from the Google Play store.

I would like it to be able to do this using my existing home internet router, without the need for a further monthly subscription to some phone-based data plan. (I only intend to use this phone from home. I am with Sky Broadband for my home router.)

I would like the phone to be as cheap as possible, about £50 or slightly above. (I want to make sure it is right for me before spending loads of cash.)

I would like it to be able to play simple games like checkers, chess etc.

I live in the UK, if that is important.
I bet your home router supports WiFi. It's the same wireless networking technology that laptops use to connect to the Internet. Android devices have WiFi too, you can enable it and connect to your router to use its Internet connection.