Why did President Kennedy seem cautious at first on civil rights?
The Democratic Party was losing ground in the South over the Civil Rights issue.
Kennedy belonged to the Democratic Party. "Democrat" has meant different things over the course of the nation's history; it was founded by Southerners to promote the business interests of slave-owning Southern planters, and after the Civil War it was the party that registered dissatisfaction with Northern government and business interests. In the South, there were few Black Democrats or white Republicans until the 1930s.
By 1960, the year Kennedy was elected President, Northern Democrats were the liberal wing of the party and Southern Democrats were the conservative wing--but only among white voters. Roosevelt had brought Black voters into the party, and many Southern whites shifted allegiance to the more conservative Republican party. The Democrats lost the lock they had held on the South since Jefferson's time.
Kennedy had a plainly progressive agenda with regards to Civil Rights, but knew that pursuing it ham-handedly would cost his party the South, the Presidency and both houses of Congress. Leaving all of these to the Republicans--who hadn't supported a progressive social agenda for over 50 years at that point--would have damaged the Civil Rights legislation he wanted. He proceeded with caution.
His successor, Lyndon Johnson, pursued Civil Rights legislation with a lot more vigor very early on, and white Southerners have voted Republican en masse ever since.