The main thing you can do to save on your taxes if you are married is to file jointly. You almost always pay less tax if you file jointly vs. filing separately. Instead of specific options being available if you are married filing joint, there are restrictions and limitations on what you can do if you file separately. The question What are the different tax savings options available to those who are filing jointly?" is too broad to be answered on this forum. Eligibility for most tax saving options are dependent on the specific circumstances of the taxpayer(s).
As for your questions about Investment options, those are questions better suited for an investment advisor that works with Indian citizens living in the US. Your questions are beyond the scope of our US tax expertise and could constitute providing investment advice, which we are prohibited from doing.
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