Congratulation and wishes in Hindi can be tricky for the beginners as there maybe more than one way to congratulate.

Let me show you how to congratulate in Hindi. Usually, it is quite simple to congratulate someone if the setting is informal.
You can simply say,

Congratulations!
बधाई हो!
Badhai!

or

Many many congratulations!
बहुत-बहुत बधाई!
Bahut-bahut badhai!

However, in formal setting, you can congratulate someone by saying बहुत-बहुत हार्दिक शुभकामनाएं। (bahut-bahut hardik shubhkamanyein which means many-many heartily wishes) or बहुत-बहुत बधाईयाँ। (bahut-bahut badhaiyaan which means many-many congratulations) depending on the occasion. Of course, you can simply use बधाईयाँ or शुभकामनाएं while omitting the other terms. The longer version is quite frequently used in the formal settings.

Usually, the structure or format seems to have at least 3 parts.

Part 1 (Personal pronoun + post-position को (ko – to) ) + Part 2 ( occasion + post-position पर (par – on) ) + Part 3 ( बहुत-बहुत हार्दिक शुभकामनाएं। or बहुत-बहुत बधाईयाँ।)

So let me show you these 3 parts with some examples and other details.

Part 1. Personal pronoun + post-position को (ko – to)

To you (formal) – आप को – aap ko
To you (informal) – तुम को – tum ko
To you all – आप सब को – aap sab ko
To all – सभी को – sabhi ko

Part 2. Occasion + post-position पर (par – on)

Occasion can be a milestone, achievement, festival or day etc.

Milestone (Note: use बहुत-बहुत शुभकामनाएं।)
On birthday – जन्मदिन पर – Janamdin par
On wedding – विवाह पर – Vivah par

Achievement (Note: use बहुत-बहुत बधाई।)
On promotion – पदोन्नति पर – Padonnati par
On success – सफलता पर – Success par

Name of festival/day + post-position पर (par – On) (Note: use बहुत-बहुत हार्दिक शुभकामनाएं।)
On Diwali – दिवाली पर – Diwali par
On Holi – होली पर – Holi par
On Merry Christmas – बड़े दिन पर – Bade din par (Note: Christmas is called Bada Din in Hindi)
On Easter – ईस्टर पर – Easter par
On Eid – ईद पर – Eid par
On New Year – नव वर्ष पर – Nav varsh par

Part 3. Congratulations/Wishes
बहुत-बहुत हार्दिक शुभकामनाएं। (bahut-bahut hardik shubhkamanyein – many-many heartily wishes)

or

बहुत-बहुत बधाईयाँ। (bahut-bahut badhaiyaan – many-many congratulations)

or simply, बधाईयाँ (badhaiyaan – congratulations) or शुभकामनाएं (shubhkamanyein – wishes)

Now, if we want to congratulate, just fit the pieces together (occasion, festival) into the format mentioned above.
Let me give you an example on how to wish someone on his/her birthday:

English: Many-many heartily wishes on your birthday!
Hindi: आप के जन्मदिन पर बहुत-बहुत हार्दिक शुभकामनाएं!
Romanized Hindi: Aap ke janmdin par bahut-bahut hardik shubhkamanayein!