British East India company's early operations in India in 1600s

French coat of Arms. credit: wikipedia
Dutch East India Co. ships.
Indian trade links with Europe began through sea route only after the arrival of Vasco da Gama in Calicut, India on May 20, 1498. Surely the new sea route gave an impetus to the European traders because new mercantile opportunities were open for them in unexplored Indian subcontinent and adjacent countries. Conflicts of interest and competition created rivalry among the them to establish their trade and control.
Sir James Lancaster commanded first EIC voyage -1601. credit: en
Anticipating a healthy  future mercantile trades in SE Asia, in 1601 the East India Company was chartered, and the English began their first inroads into the Indian Ocean. At first the British were little interested in India, but rather, like the Portuguese and Dutch before them, were interested in spices of Spice Islands. Unable to dislodge the Dutch from Spice Islands in 1610, the British chased away a Portuguese naval squadron, and the East India Company created its own outpost at Surat. In 1608, the English East India Company established a settlement at Surat (now in the state of Gujarat), and this became the company's first headquarters town. This small outpost marked the beginning of a remarkable presence that would last over 300 years and eventually dominate the entire subcontinent and later the globe.
Dutch East India co. credit:wikipedia
In the mid-19th century, the East India Company had become the paramount political and military power in South Asia, its territory held in trust for the British Crown. Slowly and steadily the company's' expansion was progressing steadily. The presidency town of Calcutta was established in 1690 at Fort William. Madras Presidency was  established in 1640. Bombay Presidency -  East India Company's headquarters moved from Surat to Bombay (Mumbai) in 1687. 
Flags of East India co.
The company rented a trading outpost in Madras in 1639. Bombay, which was ceded to the British Crown by Portugal as part of the wedding dowry of Catherine of Braganza in 1661, was in turn granted to the East India Company to be held in trust for the Crown.
British East India co currency/bill  credit.
EIC, which was incorporated on 31 December, 1600, established trade relations with Indian rulers in Masulipatam on the east coast in 1611 and a trading post in Surat on the west coast in 1612. As a result of English failure to compete with the Dutch in the Spice Islands, they turned instead to India. In 1614, Sir Thomas Roe as instructed by King James I, visited the court of Jahangir, the Mogul emperor of Hindustan and succeeded in arranging a commercial treaty and to secure for the company sites for commercial agencies, -"factories" as they were called. East India Company set up factories at Ahmedabad, Broach and Agra. In 1640 East India Company established an outpost at Madras. In 1661 the company obtained Bombay from Charles II and converted it into a flourishing center of trade by 1668. English settlements came up in Orissa and Bengal. In 1633, in the Mahanadi delta of Hariharpur at Balasore in Orissa, factories were set up.
East India co. credit:wikipedia
 Meanwhile, in eastern India, after obtaining permission from the Mogul Emperor Shah Jahan to trade with Bengal, the Company established its first factory at Hoogly in 1640. Almost a half-century later, after Emperor Aurangzeb forced the Company out of Hooghly, Calcutta was founded by Job Charnock in 1686. An English factory was set up in 1651 at Hugli (also Hoogley). In 1690 Job Charnock established a factory and in 1698  it was fortified and called Fort William. The villages of Sutanati, Kalikata and Gobindpore were developed into a single area called Calcutta. Calcutta later became a major trading center for East India Company.
Ft. St.George, Chennapatnam 1639 credit: en wikipedia. org
Red Draggon British ship fought against Portuguese  wikipedia
Once in the Indian subcontinent, the British began to compete with the Portuguese, the Dutch, and the French. Through a combination of outright competition and clever alliances with local princes and choice of locations backed by well-organized military power, the East India Company gained control of all European trade in India by 1769. As for the Dutch their focus was more on trades than on territories. So, the Dutch were not a big threat to  the British. In 1672 the French established themselves at Pondicherry and it started a rivalry between the British and French for control of Indian trade.Besides the intention of the French was to establish a French empire and at the same time build a strong trading base in the vast Indian subcontinent.

In 1659,the British defeated the Dutch settlements  in Chinsura and captured all the settlements in South India. Only in the 1700s the British had a tough time with the French competitors after they had a strong foothold in Pondicherry.The twenty years' struggle between the French and the English ended in 1763. The settlement of Pondicherry and a few other places were restored to the French, but the English remained a force to reckon with  in Southern India.