Can we use the same technology we use for radios to create a receiver which would catch Internet data?
No, radio receivers only receive radio data.
However, Wi-Fi transmits Internet data over the radio.
So can we build a radio receiver that will receive Wi-Fi data? Well, yeah. They're called Wi-Fi modules. Your computer has one, how did you think it received Wi-Fi data?
Can we build a receiver which would be connected directly to the satellites?
Do you have a long enough cable?
I'm not really sure what you're talking about, anyway. There are no satellites involved. Unless you have satellite Internet which means your Internet goes through a satellite. You'll know if you have that.
You could receive satellite Internet signals meant for other people who have satellite Internet, but they should be encrypted so there's no real point. (I don't know if they are encrypted - humans like to cut corners)
And if those are not possible when you turn on the WiFi and start browsing your computer retrieves information in the form of EM waves. Can we build something that could collect these information?
Yes, it's called your computer. Specifically the Wi-Fi module (chip/card/whatever) in your computer.