If you lost apps by syncing iPhone to iTunes on a new computer, don’t panic. You could get them back.
You’ll need to authorize the new computer to your iTunes account. Then you can re-download any applications you purchased.
As for connecting to the new computer and syncing, you may get a warning about it, but to my knowledge the settings (mail, calendar, etc) are from the computer to the phone, not the other way around. You may need to create the calendar and email accounts on the computer first, …

There are a couple of ways to get into recovery mode:

Make sure that your phone is off. If your phone is working normally, just hold down Power for five seconds and then slide to turn off. if your device can’t boot or is hung, you’ll turn it off by holding Power and home until it powers off.
Hold down the home button.
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Connect it to your computer via USB.
Keep holding the home button until your phone displays a screen similar to the one below, then let go.
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If later you decide to …

Ordinarily, if you want to reinstall OS X on your iPhone, you just plug it into your computer and click “Restore” in iTunes.
However, restoring works only if iTunes recognizes your iPhone. if that’s not happening, then recovery mode is the answer. Recovery mode lets iTunes know that the iPhone is ready to be restored. Restoring your iphone puts it into an almost completely factory-fresh state that includes the erasure of all data, including songs, videos, contacts, photos, calendar information, and any other data that was added to your iphone …

There are times you may want to enter DFU Mode, here are the basic steps to get into DFU mode. You have to get the timing right.
1. Turn off your phone (see figure below).
2. Hold down both the Home and Power buttons for exactly 10 seconds, then release the Power button and keep holding the Home button (see figure below).
3. After another 10 seconds, let go of the Home button as well (see figure below).
In DFU mode, there should be nothing on the iPhone screen except backlit blackness, as …

There are many articles online talk about iPhone DFU mode. Many of them are hard to understand, some are even worse – give you the wrong definition. Here I feel I should explain to those of you who wants to know what DFU mode is and how it works.
In the process of hacking your phone, you may get into a state where you just can’t get the OS to boot. Or you may get errors when attempting to restore your device. Or you may be trying to
unlock your phone …