Right, so popping off about Final Fantasy aside, let's get serious for a minute.
I just dropped this post in StraightShooter's profile and it should help you out as well:
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For your Dragon Quest Adventures:
1-3 and 4-6 are both separate trilogies containing very similar locations and themes. The first three are good games but are definitely older so your mileage may vary depending on tastes. 4,5,6 in particular are very good and depending who you ask, 5 is one of the best JRPGs around. DQ7 is great and a very, very long game even by JRPG standards. DQ8 is also very good and brings in voice acting and 3D graphics; many consider it the best of the bunch. DQ9 is very popular but it is also one of the easiest. DQ10 is a Japan-only MMO so I hope you speak Moon. And DQ11 is the newest and it is shockingly good, they haven't lost their touch.
https://youtube.com/watch?v=NdFgFABSkhQ
4, 5, 6, 7, 8 and 9 are all available on the DS/3DS either in "remake" form or as an original release. Some of them are pricey to acquire nowadays but the remakes are mostly solid and worth your time.
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So there's that summary. Like you say, the original trilogy is dated, though they are definitely a lot of fun if you can put yourself into the frame of mind of a early-gen JRPG. I would strongly recommend playing IV, V and VI in order. You won't be disappointed and you will be able to see how the developers of Dragon Quest utilize different presentation styles to convey the story within their game.
Even though I think DQV is the best of the series, I personally enjoy Dragon Quest VII the most because it is packed with content and I have a thing for games that just keep on going. To that end I can also easily recommend the newest entry, Dragon Quest XI.
The only one I can't talk about is Dragon Quest X because it was region-exclusive and I don't speak Japanese.