I would assume the people shouting "Segregation forever" didn't take down their "Whites only" signs the day after the Civil Rights Act or Voting Rights Act were signed. How did the implementation of those laws work? How did the segregationists resist?

Your assumption is correct. There are two major patterns of resistance to and implementation of civil rights legislation that played out through the 60's and 70's.

The first pattern, the legalistic pattern, begins with the Heart of Atlanta Motel, Inc. v. United States, 379 U.S. 241 (1964) case. The Civil Rights Act of 1964 was signed on July 2, 1964, and oral arguments for this case, which hinged on violation of the act, were started Oct. 15, 1964. This case was, for both sides of the segregation/desegregation "debate", a test, which the segregationists lost. It continues with, well, a boatload of court cases for the Voting Rights Act (1965), but firstly South Carolina v. Katzenbach, 383 U.S. 301 (1966) quickly followed by Katzenbach v. Morgan, 384 U.S. 641 (1966). Likewise for the Civil Rights Act of 1968 (The Indian Civil Rights Act, the Fair Housing Act, and the Anti-Riot Act), whose test case was Jones v. Alfred H. Mayer Co., 392 U.S. 409 (1968). (I would have used the LOC, but all the download links are broken. So Justia it is.)

(Not strictly relevant, but Jones v. Alfred H. Mayer Co. has this quote:

[Discrimination on the basis of race], this Court concluded, was prohibited by Page 392 U. S. 419 § 1982 [the US Code portion cited by the segregationists]. To suggest otherwise, the Court said, "is to reject the plain meaning of language."

Which is Supreme Court for "Are you stupid?")

While all four cases above have a lot of dense legalese, they're honestly fairly forthright in refusing the arguments offered by the segregationists, and occasionally go above and beyond to deflate arguments offered, which has the added benefit of providing a strong guideline for future lawsuits and the lower courts. (Reading the Syllabus and Held sections under "Case" is probably worthwhile if you're interested in this topic. They're about a paragraph each per lawsuit.) The ACLU and NAACP share in this particular pattern, as they assisted/put forward several of these cases. Congress acted fairly decisively during these years, led at least in part by LBJ's efforts, and directed by patterns of protest throughout the decade.

The second pattern, the pattern of violence, also must be considered.

White individuals, generally authority figures, send people to the hospital or to the morgue (or hospital and then morgue). Those people are largely, but not always, black. (Malcolm X was almost certainly not an example of this, as the three men who assassinated him were members of the Nation of Islam, but conspiracy theories persist.) I will remind everyone that the march from Selma to Montgomery took three separate tries and precipitated the creation of the Voting Rights Act in the process, and even when successful, a white volunteer, Viola Liuzzo, was shot and killed by four Klansmen for driving a black man home afterwards. (There's a whole other side of that story to do with the FBI, which spends a significant amount of these years playing both sides of the segregationist/desegregationist struggle--bringing court cases that otherwise would never have seen the light of day, but also running substantial smear campaigns and sabotage efforts against civil rights leaders and movements for their 'subversive' aspects. Today, the failures are the most obvious, but they also are probably the only reason Liuzzo's murder and the below case got trials.) Repeat similarly for the Chicago Freedom Movement (as well as the various anti-establishment riots in Northern cities during 1964-7, and the riots after Martin Luther King, Jr.'s assassination) and the Civil Rights Act of 1968. (For more on these riots, see the Kerner Report, generated to attempt to solve these issues.)

I can't list them all--even to list a representative sample would be beyond the scope of this reply; so, let us consider an example case: The Murders of Chaney, Goodman, and Schwermer. On June 21, 1964, just after the passage of the Civil Rights Act, two workers for the Congress of Racial Equality (CORE) and a volunteer disappeared outside of Philadelphia, Mississippi, along with their car. All three had been working on the "Freedom Summer" initiative, which attempted to register and educate black voters in Mississippi (and, eventually, the South as a whole). The KKK had been increasing in numbers after the sit-ins (especially Greensboro) of the early 1960's, and had staged a simultaneous cross burning (61 total) earlier that year in the state. Chaney, Goodman and Schwermer were arrested for speeding by the deputy, Cecil Price, at 4 pm, who held them for six hours (despite that not being standard practice) in the Philadelphia jail. CORE called jails--policy, when an organizer went missing--and, despite a call recorded to Philadelphia police at 5:30 pm that day, were not informed of the three's location. At 10 pm, they left the jail, followed by two cars of KKK members that Deputy Price had contacted via the Baptist minister/KKK leader Edgar Ray Killen. They were murdered, their bodies hidden in a dam, and their car burned. The FBI opened a kidnapping case the next day from their office in New Orleans, and on June 23, found their smoldering car hidden in a forest. In their search for the bodies, they found eight other dead men, all African-American, two known missing college students and another wearing a CORE t-shirt. The other five appear to have been categorized as various John Doe cold cases. After a month and a half, on August 4, a highway patrol officer gave a tip which allowed the FBI to discover Chaney, Goodman, and Schwermer's bodies. It is at this point I must mention that Goodman and Schwermer were white.