Judges rarely grant a defendants request for change of a public defender or other court-appointed lawyer. Disagreements between government-paid lawyers and defendants over strategic decisions are common and rarely are cause for a change of counsel. However, if in a serious case a schism between a lawyer and a defendant is so severe that a professional relationship is impossible, a judge may grant a defendants request for new counsel. Incompetence of counsel is another possible basis of a change of counsel, but one that rarely arises.
On the other hand, defendants who hire their own attorneys have the right to fire them at any time, without court approval. A defendant doesn't have to show "good cause" or justify the firing. After firing a lawyer, a defendant can hire another lawyer or perhaps even represent herself. Of course, changing lawyers will probably be costly. In addition to paying the new lawyer, the defendant will have to pay the original lawyer whatever portion of the fee the original lawyer has earned.
Your right to change lawyers is limited by the prosecutor's right to keep cases moving on schedule. If you want to change attorneys on the eve of trial, for example, your new attorney is likely to agree to represent you only if the trial is delayed so the new attorney can prepare. The prosecutor may oppose the delay, possibly because witnesses won't be available to testify later on. In these circumstances, the judge is likely to deny your request to change lawyers.
For answers to your questions about every part of a criminal case, get The Criminal Law Handbook: Know Your Rights, Survive the System, by Paul Bergman and Sara Berman (Nolo). If you are looking for a criminal defense attorney near you, check out Nolo's trusted Lawyer Directory.