Option 1: You can open a personal account easily with many banks if you come to the US
Option 2: It is theoretically possible to open personal accounts remotely without visiting the US but it's difficult. You can try with a bank that has a branch in your city in India and don't take no for an answer.
For a US business it's pretty easy to set up an account, I did one yesterday with Chase bank (I'm a non resident of the US but am in New York for 2 weeks). Once again much easier in person. It's theoretically possible to do it without visiting the US but very difficult.
For a foreign business it's theoretically possible but requires a lot of effort and asking many banks. Most will say it's not possible, but I know it is and have more than one friend that did it.