Regarding the Careless Driving and Failure to Notify Police of an Accident, both are Class 2 misdemeanors. Class 2 misdemeanors are punishable by: 3 months to 1 year in county jail and a fine between $250 to $1000.
Because these offenses are punishable by jail time, things occur differently than in normal traffic court. On the first court appearance, you may be informed that, because the consequences of these tickets are severe, you will require an attorney. The judge may then inquire into your wishes regarding the attorney. You will have two options: you can inform the judge that you would like to hire an attorney or you can ask (if you qualify) for a public defender.
If you ask for a public defender, the judge will inquire into your financial status. They may ask about your assets and liabilities (or they may make you fill a form out instead of answering out loud). If the Court, based on the information you provided, determines that you qualify for a public defender, then he will appoint them. If not, he will inform you of this and you will have to retain an attorney on your own.
As you can see, everything I just mentioned only deals with just having representation. Your tickets/charges are not even brought up yet. As you can see, because of the nature of your charges, this will not be a "one and done" court appearance. It will involve a few court appearances.