You don't even have to turn the machine off. Just power the monitor up and plug it in, and the machine will recognize it within 10 secs or so.
jrhmobile gave good info. I agree with him, nearly any store will help you find the monitor with the right connectors or adapters.
I will disagree a little with the size suggestion. Personally, I like large monitors, because I put them at the back of a fairly deep desk. My eyes are literally 4-5 feet from the monitor, depending on whether or not I'm leaning back. But, my mom's desk is a more normal size, so her monitor is probably only about 2 1/2-3 feet away. A 24" monitor looks huge from that distance. That would save you a few bucks. Up to you to decide which is more comfortable for you.
One techie thing to be aware of. You have the option to Duplicate the displays, or Extend the displays. Duplicate means just what it says. It will show the same information on both monitors. Extend the Displays will treat it as a separate space to display extra windows.
Again, it's a personal preference thing. Most people opt to Extend the Displays. It's amazing how convenient it is to have windows open on different displays that don't overlap each other and get in each other's way. And, even though you're having trouble reading the laptop screen, you may want to do this, anyway. Even if you always move your windows over to the large display, by Extending the Displays, you get the full advantage of the extra pixels available on the larger screen. Otherwise, it'll be limited to the number of pixels on the laptop screen.
You can find this option in the Display Settings. Start | Settings | System | Display, down near the bottom, below the Multiple Displays heading.
The other thing to do on this settings screen is, at the top, you'll tell the computer how you have the screens physically arranged on your desk. The computer can't see where you put the monitor. So, you use the rectangles numbered 1 and 2 at the top to arrange them to fit how you have them on your desk. If the big screen is to the left of your laptop, drag the bigger rectangle to the left of the smaller one, etc. And, if the big one has a tall stand and the laptop is flat on the desk, then you can drag the big one up, so that the bottom of the big rectangle is arranged similar to what you see on your desk. That way, when you drag your mouse from one to the other, it'll line up. Otherwise, when you drag the mouse from one to the other it'll jump up or down a few inches, depending on which way you're going. That can be disorienting. You can drag a little and test over and over again, until you get it where you like it. Due to difference in pixel sizes, you probably can't get it perfect, so don't let that frustrate you. Just move the mouse near the bottom. Then halfway up. Then near the top. If you can get the mouse pretty close when it's half to three quarters of the way up the lower screen, you'll probably find that the most comfortable.