Helper verbs after て: done in advance, tried out, regretted, left standing — and てある vs ておく, ていく vs てくる.
〜ておく V-te + おく (casual: V-te → とく)
Verb (te-form) + おく. Means to do something in advance as preparation, or to leave something as it is. Casual form is 〜とく.
パーティーの前に飲み物を買っておく。 I'll buy drinks before the party.
〜てある Transitive V-te + ある (object marked with が)
Transitive verb (te-form) + ある. Describes a state that results from someone's deliberate action, e.g. 'the window has been opened'.
机の上に手紙が置いてある。 A letter has been placed on the desk.
〜てみる V-te + みる
Verb (te-form) + みる. Means to try doing something to see what it is like or how it turns out.
この靴をはいてみてもいいですか。 May I try these shoes on?
〜てしまう / 〜ちゃう V-te + しまう / V-te(ちゃ・じゃ) + う
Verb (te-form) + しまう. Shows an action is completed, or expresses regret that something happened. ちゃう/じゃう is the casual form.
宿題は全部やってしまった。 I finished all my homework.
〜ていく / 〜てくる V-te + いく / V-te + くる
Verb (te-form) + いく/くる. ていく shows change continuing into the future; てくる shows change up to now, or something starting or coming toward the speaker.
これから寒くなっていくだろう。 It will probably get colder from now on.