Don't forget to go see him tomorrow.

Literal

Tomorrow, forget-not-ずに him [dative-に] meet [purpose-に] go.

忘れずに is a combined form: 忘れる ('forget') + ~ずに ('without X-ing'), giving 'without forgetting.' The ~ずに form is a slightly more formal/literary variant of ~ないで. Then 会いに行く is the purpose-に pattern. Chained together: 'without forgetting, go in order to meet him.'