Step 1: Identify the year of the First Battle of Tarain.
The First Battle of Tarain was fought in the year 1191 AD.
It was fought between Prithviraj Chauhan and Muhammad Ghori.
Hence, this battle is the earliest among the given options.
Step 2: Identify the year of the Battle of Chausa.
The Battle of Chausa was fought in the year 1539 AD.
It was fought between Humayun and Sher Shah Suri.
This battle comes after the First Battle of Tarain.
Step 3: Identify the year of the Battle of Haldighati.
The Battle of Haldighati was fought in the year 1576 AD.
It was fought between Maharana Pratap and the Mughal army led by Man Singh.
This battle occurred after the Battle of Chausa.
Step 4: Identify the year of the Battle of Plassey.
The Battle of Plassey was fought in the year 1757 AD.
It was fought between the British East India Company and Siraj-ud-Daulah.
This battle comes after the Battle of Haldighati.
Step 5: Identify the year of the First Battle of Mysore.
The First Battle of Mysore took place in the year 1767 AD.
It was fought between the Kingdom of Mysore and the British East India Company.
This is the latest battle among the given options.
Step 6: Arrange the battles in chronological order.
1191 AD $\rightarrow$ First Battle of Tarain (D)
1539 AD $\rightarrow$ Battle of Chausa (B)
1576 AD $\rightarrow$ Battle of Haldighati (A)
1757 AD $\rightarrow$ Battle of Plassey (E)
1767 AD $\rightarrow$ First Battle of Mysore (C)
Step 7: Final conclusion.
Therefore, the correct chronological order is D, B, A, E, C.
Hence, the correct answer is option (D).