ながら:
Conducting two actions simultaneously: 母は鼻歌を歌いながら夕飯を用意している - My mother is humming while preparing dinner.
Appearance or manner: 被害者は涙ながらに事件の状況を語った。In tears, the victim related the circumstances of the incident.
Adversative connective: 私というものがありながら、家に帰って来やしない。Although he as someone like me, he doesn't come home.
がてら: Combining two actions. Doing one makes it so you can do the other.
京都においでの節は、お遊びがてら是非私のところへもお立ち寄りください。While you're in Kyoto, please do stop by!
買い物がてら、その辺をぶらぶらしない。While we're shopping, let's take a stroll about.
つつ: Much in the same vein as ～ながら, but mostly written language.
