DQ is one of those series that it doesn’t matter which one you start with. But I’ll give you the low down on each one:
I, II, and III are commonly called the Erdrick Trilogy, since they deal with the Legendary Hero Erdrick (called Loto or Roto in Japan) and his descendants. It’s commonly agreed upon that the NES versions haven’t aged very well, and later ports (for the Game Boy Color, Super Famicom, Wii, and App Store) improve upon them anyway, so I’d go for those. Sadly, only the NES and Game Boy Color versions have made it outside of Japan, though there’s still hope (however faint) for the Mobile version. It’s probably easiest to emulate them, and for that I’d recommend a Fan Translation of the Super Famicom versions. III is the best of the bunch with its highly customizable party.
IV, V, and VI are called the Zenithian trilogy. They are not as closely linked as I – III, but they do share a superior race of beings called Zenithians. It doesn’t matter what order you play these in. It’s a lot easier to track down a legal copy of these games than the Erdrick trilogy, since they were all released for DS a few years back. It’s especially recommended you play V. It’s frequently lauded as the best in the series. (Don’t miss out on IV or VI though!)
The games started getting standalone stories with VII. This game is huge. Howlongtobeat.com currently has this title sitting at 104 hours for the main story alone. It recently got a 3DS remake… which is currently Japan-only. Good luck finding the PS1 version for a good price…
VIII is probably the easiest to get right now. It just got a Mobile version released over here! You can get the links at the top of the page. However, you might prefer the PS2 version, which has higher quality music and voice acting. Whatever you decide, don’t let this masterpiece slip by!
IX for the DS is the one that I am most familiar with, since it’s the only one I’ve beaten. It has a strong multiplayer focus, and brings back the highly customizable party from III. The fact that Nintendo Wi-Fi Connection shut down recently means that you might be locked out from a lot of post-game quests, though. There might be a workaround though.
X is an MMO for Wii, Wii U, PC, and Android, with a 3DS version on the way. It’s also currently Japan only, with no sign of localization.
You might also enjoy some spinoffs like the Dragon Quest Monsters series or Rocket Slime. You can also see Dragon Quest characters in Fortune Street.