My company is increasing in size, and our websites are picking up more and more traffic. We're currently using some low-cost shared-hosting, but have been experiencing downtime due to too many concurrent connections (limited to 200), enter Virtual Dedicated Hosting.
I have some experience with Linux, but limited to running basic commands to move, edit, delete files and directories. So, being a less-experienced user, I'm a bit shaken at the thought that everything will be on my shoulders. We're going to be running Apache, PHP, and MySQL.
What types of things should I be focusing on with regards to reading and learning. Any major errors new sysadmin's make when acquiring their own server?
Okay, being realistic here...
You're asking for trouble to make a quick jump from shared hosting to a dedicated, self-managed VPS, at your experience level. The best thing that you can do is sell your management on a two-step upgrade process.
Step 1 - migrate to better shared hosting. Yes, tell them they need to actually spend some money. Do this for 6 to 12 months, while you begin step 2.
Step 2 - migrate to VPS hosting. The majority of this time you'll spend learning what you need to learn (and it's alot) to effectively admin a server.
This gives you time to research VPS providers, pick a Linux distro, practice install/configure/backup/restore, mange security, learn routing/firewalls/ssh/etc, all on your own private little test network that consists of a spare PC or two, along with a good VM manager such as VirtualBox.
I think that you'll be able to learn all of this stuff, but it's unrealistic for your management to dump it in your lap like this. Adminning a server is not like using shared hosting, where the most you'll need to do is upload your web files via ftp.
Find a service that includes Plesk with their VPS package. Plesk is a web-based server management GUI. If you're even a semi-competent sysadmin, you'll be able to easily manage the everyday tasks using it.
But if you're serious about broadening your skillset, I would recommend that you find an old computer in a storage closet at work and install Linux on it (the same distro as your VPS). Use it as your in-house development/testing server and learn how to admin it via SSH.
Whoa, you're in for quite some work there if you are to manage the server yourself. You might want to take a distribution that eases your task if you can afford the choice. Distribution like Ubuntu are good for learning Linux over time, they are easier to configure and operate than some others and still can let you go deeper if you are not afraid of command line shells.
You could also look into some webtools to manage your server (like Webmin), but those will require LAMP (Linux Apache MySQL and PHP) to work in the first place.
I heavily recommend getting one of the For Dummy book about administrating a Linux server one of those days if you are to be so involved with a server :)
Good luck to you! You'll see, after a few years, it comes easily :P
Sounds like your on a tight timeline, so you should probably get some help with this one. If you do choose to go it alone you should know that most VPS accounts are a bit stingy on RAM. It's common to use a web server besides apache for this reason. Lighttpd, Nginx, etc are good choices.
Stick with Ubuntu or another distro, they will have all the packages you need available in the default repo. I run CentOS but I end up adding additional repositories and setting repo priorities. Since you aren't all that familiar with such and are on a timeline you should probably stick with a debian based distro at first.
Work on getting the site up under an alternative domain such as dev.domain.com so you can test it before the switch date. Only change one thing at a time and test it fully. Don't add APC, additional mysql options, ssl, all at once.
I honestly believe that this could be a great learning experience in the long run but disastrous in the short. Everything I know about hosting is due to three years trial and error..not three weeks or so. Good luck.