You should discuss the filing of a notice of appeal with the public defender who handled the case. It is very important that this be done immediately, because a notice of appeal must be filed within 10 days of final judgment. It should be noted that your appellate public defender may be different than your trial public defender. Even if you are dissatisfied with the public defender's office at this point, I recommend that you do not try to proceed on appeal without an attorney. This is because there are specific briefing rules that must be followed in the appellate courts in Missouri. Failure to comply with the rules may result in the dismissal of your appeal. If you do wish to proceed on appeal without an attorney because you cannot afford one, you need to file a motion to proceed in forma pauperis, which, if granted, will permit you to file your appeal without the payment of filing fees.
If you decide to proceed without an attorney, you should access the appellate court's website to view copies of the Court Rules of Appellate Procedure and the Local Rules of Procedure. I believe Springfield, Missouri, is the home of the Southern District Court of Appeals. Also, the Eastern District publishes an "ABC's of Appellate Practice" that you can download. Here is a link: http://www.courts.mo.gov/appellate/index.nsf/9dfd92495509f13f8625681f0079fd11/e9350c5fcbe00f72862568b0006e8125?OpenDocument.