Hi ! I am very new to j-drama. I have seen itazura na kiss 1&2 and Hana yori dango, both of which I absolutely loved. I am going out of town and on a long road trip and would like some happy ending romance j- dramas to help me on my trip. Please suggest titles that are romantic, happy ending, not older than 2000's ,and j- drama only. Thank you!

I love Hana yori dango and Itazura na Kiss too. *.*

These are some dramas with romance that I really like with happy endings:
- Nagareboshi (drama and romance)
- Innocent Love (drama and romance)
- Mei-chan no Shitsuji (comedy and romance)
- Proposal Daisakusen (romance, comedy and fantasy)

These are also alright and fulfill your requirements if you want to give them a try (not my favorites):
- Hotaru no Hikari (comedy and romance)
- Shokojo Seira (drama, school and romance, based on the novel A little Princess by Frances Hodgson Burnett)
- Celeb to Binbo Taro (comedy and romance)
- Juunen Saki mo Kimi ni Koishite (romance and fiction)
- Hotelier (romance and business)

Good luck! :D
I would recommend:
Rich Man Poor Woman
Love Shuffle 
Both dramas have one of the actors from Hana Yori Dango :)
J-dramas are my first love, here are my favorites:

1. Yamato Nadeshiko Shichi Henge (my first asian drama, remained one of my favorites)
2. Hana Kimi (a classic in Japan)
3. Proposal Daisakusen (really cute romance)
4. Last Cinderella (I loved this one so freaking much)
5. Tatta Hitotsu no Koi (one of the best romantic dramas out there)
6. Love Shuffle (comedy, romance and life genres)
Pride, Tatta Hitotsu no Koi, Orange Days, Nodame Cantabile, Love Shuffle all of which I loved. Watch Hana Kimi, Yamada kun to nananin no majo - more comedy than romance but goofy fun.

You can also try Buzzer Beat and Rich Man, Poor Woman.