N1
～てやる
Just wait and see, Just you wait, To do a favor, For you, Definitely, For sure
今夜は俺が奢ってやるから、どんどん好きなもん頼めばいいからな。
I’m going to pay for you, so don’t hesitate and order whatever you want.
Explanation
「～てやる」 has 2 uses
1Threat or vow
A threat to a rival, or a vow to show them: 見返してやる, ぶっ飛ばしてやる, 合格してやる.
絶対綺麗になって見返してやる！
I am going to become pretty and prove them wrong, just wait!
こうなったら勝ってやるしかないだろ。
Well, in a situation like this there is no other choice but to win!
2I'll do it for you
Doing something for a friend, someone younger or an animal, in a big-brother tone: 奢ってやる, 連れていってやる.
今夜は俺が奢ってやるから、どんどん好きなもん頼めばいいからな。
I’m going to pay for you, so don’t hesitate and order whatever you want.
いいよいいよ、俺がやってやるからお前は見とけ。
Don’t worry about it, I’ll do it for you, so just watch.