I would say that if you are looking for some songs that most Hawaiians (who are into traditional Hawaiian music) would recognize I would start with:
Noho Paipai
Papalina Lahilahi
He U`i
Ulupalakua
Wahine Ilikea
I Kona
Hi`ilawe
Ahulili
Henehene Kou `Aka
Ka Uluwehi O Ke Kai
Those are just off the top of my head. But most of those songs will come up at a Kanikapila and those are all in heavy rotation on Hawaiian music radio stations here.
Some of my favorites to play
Kimo Henderson Hula
Makee Ailana
Latitu
Ipo Lei Momi
Ka Pua U`i
Hanohano Ka Lei Pikake
Mahina O Hoku
I can go on and on...
I also agree about the Hawaiian Style Ukulele books with the CDs, I did buy the first one, they give you an attainable target to go for if you do not have familiarity with the songs.
He Mele Aloha is a must have if you wish to pursue Hawaiian music. Sometimes you may need to change the key to suit your voice, and remember that He Mele Aloha is designed to give you the basic framework of the song. When you hear Hawaiian groups play those same songs on the radio they may be much more complex, with key changes, extra chords, different arrangements, etc.
I am working on falsetto singing. I have a long way to go, but I really enjoy this music!