'Do I need to root my phone to use my phone?'
Android is a very open mobile OS, you can change all aspects ranging from homescreens to keyboards just fine without any kind of root unlock. Root is required if you wish to unlock certain permissions (e.g. adjusting the CPU speed) and/or install custom ROMs.
A custom ROM is a community-made modification of the Android OS, customizing it more in some way. Additional functionality (like joining ad-hoc wifi networks) or more sleek set-up (speed, control over battery life) can be gained. You're no longer reliant on your manufacturer or carrier to push android updates to your phone. You could for example run a modified gingerbread (Android 2.3.3) on your ZTE Blade phone instead of Android 2.1 eclair.
Root also opens the door for automatic battery modifications by automatically underclocking your CPU when your phone is idle or screen turned off. Some ROMs have kernels that support advanced 'governor' settings so you don't have to tweak with profile settings either.
'How do I get started?'
You will need to gain root access on your device and use a ROM specifically suited for your phone model.
Check the XDA dev
forums, scroll down to the forums of your device and check out the Sticky threads. Be sure to follow them with care, backing up where they ask you to back-up your original stuff and using the right instructions for your device versions.
a few basic terms:
ROM: this is the OS you'll be flashing to your device. Examples include official ROMs, Cyanogenmod etc.
Recovery Mode: when you boot your phone to recovery mode (usually volume down + boot) you hit recovery mode. Rooting your phone installs a custom recovery mode (usually either clockworkmod or amonra) from which you can flash .zip files to your phone, such as the ROM you want to install
Nandroid: in Recovery mode you can make a complete system image back-up of your phone to which you can restore to. It is recommended to _always_ do this before wiping your phone and flashing a rom. This way you can return your phone to the complete image state it was before.
Radio: The firmware of the cellular radio can also be flashed, check the ROM page for which radio version is recommended.
when flashing a ROM of a different version (custom ROM or android version) you should wipe your device in recovery mode first. As such, you'll need to make a back-up of your messages, call logs, contacts etc first. Titanium Backup pro works (but requires root first, so root first, then back-up, then flash the ROM).
Rooting has its risks! Be prepared to have to tinker with your device in case shit goes wrong.
Also, before you do anything to your phone in recovery mode (you'll probably have to go there as part of the rooting process if you decide to, as well as if you are flashing any custom ROMs) make sure your battery is charged! If you're plugged in it won't charge while in recovery; this is probably the quickest way to brick your phone.
While flashing your device it's recommended not to charge the device - it may fool battery calibration a little bit (although that can be re-calibrated later).
Note that while rooted you may not be able to receive official OTA updates, or if you can - they will make you lose root. If you install a community ROM they will most likely incorporate the official update in a newer version.
Recommended root apps:
SetCPU, Titanium Backup, ROM Manager, ADFree Android, ShootMe (Screenshots)