****WARNING! LONG POST!****
I understand that after a while, you will get to know the community and will become familiar with different members' strong points, and you may wish to PM them with specific questions. I would encourage everyone, as much as possible, to post everything right here in the forums. That way the question is already answered for the next newbie that needs help. Just something to keep in mind.
Let me see if I can spare you some time and confusion. I'm sure others could explain this is much more technical detail, but I'll try to keep it simple and approachable. This is geared toward any newbie looking to get the most out of his shiny new Android device, so I apologize if some of this sounds weird to you.
1. Bootloaders: locked and unlocked.
Bootloaders are the most low-level environment on your phone that's available to users, as far as I know. It's somewhere between BIOS and GRUB for those of us familiar with such things. (If you don't what I'm talking about, just ignore that LOL.) It allows access to
fastboot mode, which allows you to run various commands from your computer via the terminal. It also gives access to
Recovery Mode. This is also akin to the recovery mode for some desktop OSes. More to come on that. By default, any new device will come with a
locked bootloader. The technical details aren't important. What is important is that a locked bootloader will only work with stock recovery and stock firmware (ROM). To be able to flash a custom recovery, you must first unlock the bootloader.
You'll note it shows "S-OFF" (presumably "Security" though I could be wrong on that).
2. Recovery
Stock recovery allows just a few operations on your device. You can do a factory reset, flash an official ROM, and maybe a couple other things. You cannot make backups, partition an SD card, flash custom ROMs, etc. For this you need a
custom recovery. The most popular (probably because it's the most supported) is ClockworkMod Recovery, developed by Koushik Dutta, aka koush of the CyanogenMod team. There are probably others out there, but CWM should be more than sufficient for your needs. Plus, using CWM has a couple other bonuses we'll get to later.
3. Android Market (aka Gold Mine)
http://market.android.com <-- awesome for PC browsing. You can even install apps directly to your device Over the Air (OTA) directly from the web. Super sweet! All you have to do is sign into the same account in your browser as you did on your device. If you have multiple devices you can choose which device to install each app to. You can purchase apps, rate, comment, etc., all from the web Market. It's awesome, really. That being said, here are a few apps that can make life easier if you're looking to tinker with your device, or even if you just want to get the most out of it.
3.1
ROM Manager
As the name implies, this is a simple, yet very powerful and feature-rich app that helps tremendously when you want to play with custom ROMs. It allows you to flash CWM directly (after you've unlocked the bootloader), perform backups, flash ROMs, and a few other things as well. There is a premium license you can get to unlock a bit more functionality, but the free app does quite a bit already.
3.2
Root Explorer
This is a file system browser that allows you to perform simple file system actions, such as copy/paste/delete/rename files, open text files for editing, play media files, and a few others. There are free file explorers out there, but this is my personal favorite, and if you've rooted your phone, Root Explorer will be able to access and perform actions on otherwise protected parts of the system.
3.3
Swiftp FTP Server
Call me lazy, but I just don't like having to plug my device into my computer to transfer files back and forth. This is a simple FTP server that will allow you to transfer files back forth to your computer over Wifi using any FTP client.
3.4
SwiftKey X
As you've heard, this is one of the most popular "traditional" tap-to-type keyboards (and my favorite of this type). Its word prediction is downright creepy sometimes, it's that good, and its error correction is also very good. Yeah, it costs money, but I think it's worth it.
Swype
This is it. It's gotta be one of the most popular input methods available on
any platform. It's not for everybody, but I'm fairly certain you'll enjoy it. It comes pre-installed on your phone, but if you start flashing custom ROMs, you'll want to get the beta, since most custom ROMs don't (and shouldn't) have Swype included.
That's all I can think of for now. Like I said, feel free to ask questions if you don't understand something. I'd encourage you jump into an IRC channel for realtime help if you get stuck at some point in the rooting or unlocking process or something. We're always here to help too, so keep those questions coming!
P.S. Please overlook any misspellings, missing words, or grammatical errors. I tried to be careful, but I may have goofed in there somewhere.