Here are some differences between Sunni and Twelver Shi'i practices during Ramadan and Eid.

Ramadan:
* Shi'is usually fast a bit longer than Sunnis because the maghrib time is about 15 minutes later. Fajr time may vary slightly too. Usually Shi'is stop fasting slightly before fajr to be careful (the time is called imsak).
* Shi'is commemorate the main nights of laylat al-qadr on the 19th, 21st, and 23rd. There are specific du'as and acts of worship for these nights which you can find on duas.org. On the 19th to 21st, the martyrdom of Imam Ali is commemorated.
* There are also some other du'as which may be recited during the month of Ramadan which you can find on duas.org.
* Shi'is do not say taraweeh prayers in congregation; however, in some places, people get together for reciting the Qur'an so the whole Qur'an can be recited in one month. Otherwise it is good to do it individually.

Eid
* Eid is celebrated on one day rather than 3 days
* Shi'is also celebrate Eid al-Ghadir and other days such as the birthdays of the Prophet and Imams
* There are some differences in the Eid prayers including the qunut which is sometimes confusing (you can look it up)

Maybe there are some other differences which some others can mention. However most of these are for optional things (like reciting dua) and not requirements, apart from the fasting times. 

Best wishes!