What types of both legal and illegal activities did segregationists use to maintain segregation during the period 1954 to 1968?
There were many legal attempts to prevent integration. For example, after the Brown decision, school districts were known to go as far as abolisihing public schools in their area so that they could not be forced to integrate them. Governor Orval Faubus shut down the high schools in Little Rock, Arkansas in the year after the "Little Rock Nine" integrated that city. Some states passed laws that allowed integrated schools but did not fund them.
Outside the law, there was both economic coercion and violence. The White Citizens' Councils did things like ensuring that blacks who pushed for rights would lose their jobs or would be denied loans for homes or businesses. There were also famous episodes of violence. Three civil rights workers were murdered in Mississippi. Black churches were bombed. There were riots in various cities meant to prevent desegregation.