Because a first offense is not criminal, you cannot be represented by a public defender for that offense. You still have the right to retain an attorney for a first offense, though it is not required.
Because it is not a criminal offense, you do not have the right to appointed counsel on a first offense OWI; however, it is always a good idea to seek representation when you are charged with a serious offense. Even a first offense OWI is serious because it has numerous negative consequences. A validly entered guilty plea waives all issues on appeal and, therefore, you cannot "appeal" a guilty plea. One can, however, move to withdraw the guilty plea. Where the motion is made after sentencing, though, you must establish that a "manifest injustice" will occur unless you are permitted to withdraw the guilty plea.
This answer is for informational purposes only. By answering this question, no attorney/client relationship is created. Although the legal information is accurate, it may not be appropriate for your situation. The best way to handle any legal problem is to seek the advice of an attorney.
I defer to my Wisconsin colleagues to discuss the appellate process in your home state.
It is not required that you have an attorney, as the previous answers state. However, I cannot think of one reason why you would not want to be represented. There are many fine OUI attorneys practicing in WI who post here on Avvo. You can also use the "Find a Lawyer" function on this website.
Best of luck to you.
If your question is geared towards a potential collateral attack on your OWI 1st conviction, that's not possible as you have no constitutional right to counsel on an OWI 1st in Wisconsin.