Hi,
Buying and selling of shares will result in capita gains as per Indian Income Tax laws. However, long term capital gain will be taxed at the rate of 20%, subject to indexation benefit. Short term capital gain will be taxed as per the slab rate. However, in case the volume and frequency of trading is high, then the tax department can treat your activity as business, and the profit from such business will be taxed as per the slab rate.
Regarding the taxation in USA, I would recommend that you consult a professional who practises in US Tax laws.
Regards
CA Keerthiga Padmanabhan
M.Com., CA, LLB