ぜひ
by all means, definitely (eager wish)
ぜひ + たい / てください / てほしい / ましょう
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ぜひ adds enthusiasm to a wish or an invitation: ぜひ遊びに来てください, "you must come and visit"; 一度ぜひ京都に行きたい, "I'd really love to go to Kyoto sometime".
It goes with positive wishes and requests: たい, てください, てほしい, ましょう. It can't express a guess (use きっと) or a negative (use 絶対に).
On its own, ぜひ! is an eager reply to an invitation: 「一緒に行きませんか。」「ぜひ!」, "I'd love to!"
It's a small word that makes invitations and recommendations sound warm and sincere, so it's worth getting into your speech early.
Examples
ぜひ遊びに来てください。
You must come and visit.
一度ぜひ京都に行きたいです。
I'd really love to go to Kyoto sometime.
「一緒に行きませんか。」「ぜひ!」
"Would you like to come along?" "I'd love to!"
この本はぜひ読んでほしい。
I really want you to read this book.
ぜひまた会いましょう。
Let's definitely meet again.