I've beaten all three of the DS games (and the two for other systems) and have my own personal favorite.
My opinion is this:
1. Rune Factory 2
2. Rune Factory 1
3. Rune Factory 3
However most people in general like Rune Factory 3 the best out of the hand held Rune Factory games.
All of the Rune Factory DS games have the same gist. You play as a male with amnesia who somehow ends up in a town you've never been to before. There is some kind of plot that requires you to go into dungeons and defeat bosses. You can also farm, tame monsters, get married, attending festivals, make your own weapons/armor, etc. One of the best things about Rune Factory games is even though they have the same core elements most games have a "gimmick" of sorts. A special feature that only that game has.
Rune Factory 1:
Rune Factory 1 has some problems. Which is understandable because it was the first time they've done something like this. It also had some great features such as one of the more interesting plots and the most dungeons of any RF game so far. The game does a good job of introducing you to the Rune Factory universe. The festivals are all very Harvest Moon like and some are not very exciting. Wooing and married life also fall flat. The characters are still interesting though. They have quite a bit of dialog which is good considering there aren't many events with the characters. This game didn't really have a "gimmick". It wasn't until they made it a series that they started adding big features that were only in one game. However Rune Factory 1 does have a few features that are not in any other RF. They are not nearly as prominent as the ones in newer games though. One of the complaints is after playing newer games it is hard to go back and play Rune Factory 1 because the fighting and over all play is not as good. I never had that problem but it is a good reason to start with RF1 if you intend to get more than one game. If someone is only planning on getting one then they will often skip over this one.
Rune Factory 2:
My personal favorite out of not only the three hand held but out of all five Rune Factory games currently out. Problems that were in Rune Factory 1 were fixed in this game. There are still no events with characters but instead they make up for it by adding the Help Board feature. The Help Board is now a common element in Rune Factory games. Characters in town post requests that they need help with. If you accept a request you get a special conversation with the person before and after the request is completed. While a lot are simple there are a few that are just about getting to know characters and the actual favor is not the main focus. This is also the first Rune Factory game to have a "gimmick". The gimmick is one of the reasons this is mt favorite RF game. The second generation. You play as a man named Kyle who is the typical RF amnesia guy. But after you get married and build a school the game skips nine years ahead and you play the rest of the game as Kyle's child. Other couples in town have also gotten married and had children. In the first gen you can't enter many areas of the dungeon but once you start the second gen you can unlock new areas and reach bosses. Rune Factory 2 is the only Rune Factory game so far that has had rival couples. In the other Rune Factory games no one else can get together with any of the bachelorettes.
Rune Factory 3:
My least favorite Rune Factory game over all. I have many reasons I personally don't like it but I don't want to turn you off from getting something you may like because of my own bias towards it. Especially considering most people who play it do like it. The gimmick of this game is that you play as a character who is half monster. After the first boss you are able to transform at will into a golden wooly. Which is basically a sheep that walks on two legs. The game centers mostly around the difficulties of monsters and humans and that they don't like each other. Your main goal is to get a human hating monster settlement found in one of the dungeons to start getting along with the monster fearing human town you live in. This game also has the Help Board feature but it has been tweaked. And now instead of that being the main way to learn about the marriage candidates you also have the option of taking them out on a date to special spots in dungeons. The dungeons are far smaller in this game than they were in RF2 and there are fewer than there were in RF1. But the fighting is improved and there is a new weapon option, duel swords. Plus there is an extra dungeon you can fight in. However it is not anything plot related. There was also changes to the way farming worked such as making the soil lose quality after being used and a bag of seeds plants less crops than in past RF games.
Good luck on picking a game. Whichever one you get I hope you enjoy it.