[AT&T, Rogers, Bell, Telus Samsung Galaxy S III] mrhaley30705:
There are three parts to this thread - How to get into download mode, how to get into recovery mode, and how to solve problems connecting your Android device to your computer. I think you should be able to find where each one begins without too much trouble.

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GETTING INTO DOWNLOAD MODE

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"Just get into download mode and..."

Ever find yourself with a phone that's misbehaving, and you can't seem to figure out how to get it to download mode?

Here's a list of things to try, sorted in a "where are you now" setup:

Note: You will need to use Vol+ to select download mode after you reach the mode selection screen.

FROM A POWERED-DOWN STATE - BATTERY REMOVED

- Insert battery, then press and hold Vol-, Home button, and Power.

- Press and hold Vol-/Home/Power, Insert Battery.

- Use a USB jig (can be manufactured jig or homemade chain of resistors)

FROM CLOCKWORKMOD RECOVERY MENU

- Position the cursor (highlighted selection) on "Restart device", hold Vol-/Home, then press power

- (Where available) Use the "Reboot Download" option in the "Advanced" menu

FROM A FULLY BOOTED PHONE STATE, WITH USB DEBUGGING ENABLED

Note: you may need to have ADB installed on your computer, and/or an Android Terminal Emulator installed on your Android device.

- Connect your Android device to your computer, navigate to your ADB directory, and type "adb reboot download"

- (Where available) Use the rom's power menu option to reboot into download mode.

- From a terminal, type "su", then "reboot download"

- Download the app "quickboot", follow the app directions to reach download mode.

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GETTING INTO RECOVERY MODE (CWM RECOVERY)

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FROM A POWERED-DOWN STATE - BATTERY REMOVED

- Insert battery, then press and hold Vol+, Home button, and Power.

- Press and hold Vol+/Home/Power, Insert Battery.

FROM A FULLY BOOTED PHONE STATE, WITH USB DEBUGGING ENABLED

Note: you may need to have ADB installed on your computer, and/or an Android Terminal Emulator installed on your Android device.

- Connect your Android device to your computer, navigate to your ADB directory, and type "adb reboot recovery"

- (Where available) Use the rom's advanced power menu (long press power in android OS) option to reboot into recovery.

- From a terminal emulator, type "su", then "reboot recovery"

- Download the app "quickboot", follow the app directions to reach recovery mode.

- Download the app "Rom Manager", select "Reboot to Clockwordmod Recovery"

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UNIVERSAL FIRST STEPS FOR CONNECTION ISSUES

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So you've got Odin up, the phone's in download mode, it's connected to the computer, but Odin is saying "What phone?". You've tried jiggling the handle, checking the blinker fluid levels, and even using the board stretcher, but the Android guy is smirking at you with that smug green face.

Although it is possible that there is something else at work that's preventing your phone from connecting, I would suggest trying these "universal first steps" to see if you can resolve the connection issues:

1. Turn off both devices (phone and PC) - and I do mean full power-down. Go ahead and pull the phone's battery, if necessary.

2. Disconnect your phone from your computer at all points - USB out of phone, USB out of computer.

3. Start up your computer. Don't touch it, don't do anything to it. Let it fully boot and completely settle.

(Obviously, log in if it's necessary, but then let it finish booting before moving on to step 4.)

4. Start up Odin. Let that get completely started and settled.

(If on Vista/Win 7, make sure to "Run as Administrator" in the right-click menu.)

5. Connect your USB cable to a different port than the one you last had it connected to.

(If you're on a Windows desktop, the ports in the back of the computer are generally more stable and should be used. If you're on a laptop, use USB port #1)

Do not connect the phone to the cable yet.

6. Start your phone in download mode. Wait until you see the Android guy with the 4-5 lines of text in the upper left corner.

7. Connect your cable to the phone. Let it sit until Odin recognizes it (which should only take a couple of seconds).

If, for some reason, Odin doesn't recognize the phone at this point, here are a few more things to check/try:

- Repeat steps 1-7 above, making sure to change USB ports. As dumb as it sounds, sometimes the phone or computer just doesn't like particular USB ports.

- With everything still up, switch USB ports.

- Try a different OS (if possible) - sometimes Ubuntu will recognize what Windows won't, and vice versa.

- Try doing this on a different computer - again, different OS, different USB ports, they're all links in a chain and sometimes you'll come across a configuration that doesn't have the weak link.

- Try a different cable. I've had a case where I couldn't do any flashing with one cable, but everything was rock-solid and smooth on the same machine with just a different cable.

- Fully uninstall your drivers, reboot the computer, reinstall the drivers, power down, and then start again from step 2. Sometimes faulty drivers are the problem.

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Additional Note:

If you're moving from an AOSP-based rom (MIUI, CM10, etc) to a Touchwiz-based / Samsung Stock-based rom, remember to wipe data / factory reset prior to flashing a stock rom. You will hang on the "Samsung" logo if you don't.

Please note very carefully, though, to do the wipe in CWM recovery - not 3e recovery. If you do it in 3e recovery, you'll format the internal storage, meaning you lose all your pictures, music, etc.. (I learned this the hard way.)

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If anyone has any additional items / corrections, please reply and I'll update the OP accordingly. Thanks.