Companies like Share Khan, Karvy are some examples where you begin your career exclusively with stock market.
There are numerous private banks where you can begin your career

Dear Rahul,

Finance field is such a broad field that you need to be specific in your career choice. Introspect your interest area that where you see yourself and would enjoy working in. Some likes to work in Banks but some like to enter corporate or some enjoy working in broking houses. So, you are the best person to decide your own interest. You can enter
Banks
Financial Institutions
Brocking houses
KPOs with Financial Processes
Corporates
Financial Services Providers
NBFCs
Credit Rating Firms
Leasing Firms
Factoring and forfaiting firms
List can be huge..