Don't forget to take your umbrella with you.

Literal

Not-forget-[without-ず] umbrella [object-を] carry-go-[soft-command-なさい]

持って行く is a compound of 持つ (hold) + 行く (go) = 'take along / bring'. Its counterpart is 持って来る ('bring along / bring here'). The て connects the two verbs sequentially: 'hold it and go'.