I have a problem with Debian, Fedora and Ubuntu - they don't recognize my graphics card (an ATI Mobility Radeon HD 5870) or WiFi.
Are there any modern Linux distros which include suitable drivers for my hardware after installation?
Basically these drivers should be included in every OS you mentioned. Which Ubuntu version did you try?
Do you have a picture? Can you connect via cable?
If yes get ATI Drivers from here: http://support.amd.com/us/gpudownload/linux/Pages/radeon_linux.aspx
Realtek should also have proprietary drivers online.
Your laptop may be "too new" for current linux distributions, depending on a few factors. You can either wait a few months for the distributions to catch up (try using the latest Ubuntu, released only a few days ago, or maybe a beta of Fedora 15)...or you can attempt to use the binary drivers provided by the vendor.
The more "compatible" distributions package binary drivers. The more "free software friendly" distributions make you do it yourself.
Using the binary drivers generally leads to breakage, so see it only as a short-term solution.
I had the same problem; I tried several Debian-based distros but none would work with my Mobility Radeon 4250 HD. I was able to install openSUSE 11.4, but after installing the screen was blank on loading.
I then booted with