I got the original game free a while back and it has been on my GOG for quite awhile, but I've never played it. What is the best way to get started with this game? Do I need the original game to play Warband? Doesn't Warband improve the original game? I appreciate any suggestions.

Warband is a stand-alone game - no need to install any other titles in order to play it.

I've only played Warband but have heard it's an improvement in most ways over the original. Given that it's a sandbox-style game, not sure what you would gain by starting with the original; I don't think there's any particular story you're following or major overarching quest to solve.

The original game is NOT needed for Warband. Warband improves on some things, changes others (whether to the better is a question of taste) and adds more content (is more better? Up to you), also there are several DLCs for Warband (3 on GOG).
You can very well play the first game for a bit to try and see if you like it. My main problem with that game is the lack of viable ways to play; only a handful are enjoyable but even playing through the game four times will eat up hundreds of hours. Be warned, the tutorial stinks.
There is a demo of Warband that lets you play for a while but you will not see the mid-game nor end-game with that one. There is something special about your 150 soldiers against their 4500..............

Warband is very poorly optimized compared to the original, so if you have an older system, you may want to stick to the free version. Otherwise it's mostly an improvement. To get started, I would practice mounted combat in the tutorial until you feel comfortable with it, so you don't just get trounced repeatedly.