DQIX is very much a beginner's entry in the series. A lot of the challenge was toned down to focus on post game and multiplayer.
8 is the best game in the series in my opinion and the one I'd most readily recommend. It was (like 9) developed by Level 5 and they did a great job putting a modern touch on the series. The game has the most memorable characters in the series and a simple but effective skill system.
I really like 7, but it's over a hundred hours long and more for hardcore fans. This game is for exploration nuts. You save towns in the past, then revisit them in the future and slowly rebuild the entire world. It's tedious, but worth it if you're in to that kind of thing. If you want a more scaled down version of the same idea, I highly recommend playing Soul Blazer, Illusion of Gaia and Terranigma for the SNES instead. If you still want to play 7, I'd wait for the 3DS version (and hope we get it outside of Japan)
4 and 5 are both pretty easy to get in to and are both wonderful games. 4's got a split narrative among 8 party members who all come together eventually. It was an NES game, so it's not very deep story-wise, but the gameplay has a great difficulty curve. 5 is the second-easiest game in the series after 9 and it has a story based around a boy who grows in to a man, gets married and has kids by the end. You really get attached to him throughout the game, and it is a lot of DQ fans' favorite entry in the series. Both of these games were released on the DS, though they might be hard to find. I prefer those ports.
3 is also great. It's very open-ended in the middle portion, making it feel somewhat like the original Legend of Zelda or the second half of Final Fantasy VI. The plot is bare bones NES stuff, but in the older games all you care about is the challenge anyway. It has the same job system as DQIX (a little bit simpler) and much more challenging enemies. I recommend playing the Super Famicom port with a translation patch.
I don't care for the first two games, and 6 is a bit of a black sheep (though it's a good game). They aren't ones I'd recommend until you've played other games in the series.