Start a Case Milestone using Apex

Case c = [SELECT Id, Status, EntitlementId FROM Case WHERE Id = :caseId LIMIT 1];

// Ensure entitlement is set
if (c.EntitlementId == null) {
    c.EntitlementId = [SELECT Id FROM Entitlement WHERE Name = 'Standard Support' LIMIT 1].Id;
}

c.Status = 'New'; // Trigger milestone criteria
update c;

List milestones = [
    SELECT Id, MilestoneTypeId, CompletionDate, TargetDate, StartDate, IsCompleted
    FROM CaseMilestone
    WHERE CaseId = :c.Id
];

// Pause milestone
MilestoneUtils.pauseMilestone(c.Id, 'First Response');

// Resume milestone
MilestoneUtils.resumeMilestone(c.Id, 'First Response');

// Complete milestone
MilestoneUtils.completeMilestone(c.Id, 'First Response');