Thanks for the help so far! It has helped me think about things more clearly! 
I've been thinking, since we give the ethernet shield an IP, in what configuration must I set up the router (the one who allows the Arduino to go wireless)?
Different modes:
- Access Point: instantly adds a Wi-Fi hotspot to existing wired network ( -> not this I believe)
- Client Mode (TV/Game Console Adapter): Gives wired-only devices access to an existing Wi-Fi network ( -> Currently set up like this)
- Router Mode: Creates an instant private wireless network and share internet to multiple Wi-Fi devices, suitable for most hotel and home networks. ( -> not this I believe)
- Repeater Mode: Extends existing Wi-Fi, improving signal strenght and coverage
- Bridge Mode: Use host wireless network to establish a guest network with a different password
The thing is that I'm doubting that I should put it in Bridge mode, since the client itself is the ethernet module and not the router.
Regards,
Dries