Part 1 : Rooting
1. Download and install Samsung Kies if you have not already done so. This will install the necessary USB drivers to talk to your device.
2. Connect your Samsung Galaxy Tab 7.7 via USB and make a backup with Samsung Kies. This is in case anything goes wrong in the process, you can restore your data from the backup.
3. While your Galaxy Tab 7.7 is still connected, download root.zip and do a copy/paste via Windows Explorer under the folder GT-P6800, Card (see screenshot above). In the Android filesystem, this corresponds to the /mnt/sdcard/extStorages/SdCard folder (use this path if you had downloaded the root.zip file directly to your tablet).
4. Unplug the USB cable and turn off the tablet (hold down the Power button and select Power Off).
5. Turn on the tablet in recovery mode. This could be tricky if you have not done this before. Hold down both the Power and Volume Up buttons until the tablet turns on, and keep both buttons pressed until the recovery screen comes up. Some instructions say to release the Power button earlier when you see the device turned on but I've found it does not work half the time.
6. When in recovery mode, select install update from external storage, look for and select the root.zip file using the hardware buttons (Volume Up, Volume Down, Power).
Part 2 : Install CWM Recovery (Optional)
Installing the CWM Recovery image is an optional procedure but it had saved me a couple of times with my Galaxy S2 while I was playing with custom kernels and custom ROM's and such. This is a utility that replaces the default recovery module (when you press Power and Volume Up on boot). It allows you to make byte-for-byte backup images of your boot partition, system and user apps and data, and restores the entire system as necessary. Something like what Norton Ghost or Acronis TrueImage do for Windows computers.
To install this, you have to root your device first, i.e. Part 1 above must be completed successfully. Here are the steps :
1. Download Odin3 v1.85. This is a utility that allows you to flash custom firmware and kernels to your Android device. Simply unzip it to an empty directory. There is no installation required.
2. Download P6800_CWM_Recovery_22.214.171.124.tar and save it to an empty folder.
3. Close Samsung Kies. If you have any other application running that syncs mobile phones or other devices over USB, such as Nokia Suite, Microsoft ActiveSync, Zune, or anything similar, close it as well. This is to prevent conflicts with Odin when it is communicating with your tablet.
4. Plug in the Galaxy Tab 7.7 via the USB cable. Turn it off (hold down Power button and select Power Off).
5. Turn on the tablet in download mode. Hold down both the Power and Volume Down buttons until the tablet turns on, and keep both buttons pressed until the download screen comes up. This is different from the recovery mode in the rooting part in Part 1 which uses the Volume Up button. Some people set it in the wrong mode and nothing seems to work. Myself, I had forgotten wholesale on a few occasions to go into the download mode with the same result. If you have booted successfully into download mode, you should see some warning messages and so on. Follow the instructions until you see the waiting to download message.
6. On your PC, run odin3 v1.85.exe. Click on the PDA button and select the path to the P6800_CWM_Recovery_126.96.36.199.tar file you downloaded earlier (see screenshot above). Click the Start button, wait a few seconds for the image to transfer over, let it reboot, and that is it.
7. The next time you boot into recovery mode via the Power and Volume Up buttons, you will see the new CWM Recovery screen instead of the default recovery screen. You can then make full system backups and restores as you wish.
Disclaimer and fair warning : this is a compilation and summary of procedures that have worked for many people. It should generally be safe. But I shall not be responsible if your device is bricked or your warranty is voided, data is lost, your device goes up in smoke, etc. Like I said, do this only if you are sure you know what you are doing, and you know why you want to root your device.
This compilation of instructions is due mostly to the XDA Developers forum, Galaxy Tab 7.7 section. If you like you can go over there and say thanks to the folks who have made this possible. If you face any issues you might want to ask for help there as well.
References from XDA Developers :
1. [ROM&GUIDE] Official Firmwares SGT 7.7 P6800 + P6810 Download
2. [GUIDE] Root
3. CWM RECOVERY 188.8.131.52 [GT-P6800][3g]