How do you find a trustworthy attorney who won't take you to the cleaners and then leave you hanging with a great many unresolved issues?
Family Law Attorney
Your question is very broadly worded. On an appeal, the Court of Appeals will not hear new evidence. The scope of their review is generally limited to the record that was created in District Court. Generally speaking it is possible to file an appeal, however whether or not the result could be considered "successful" is an entirely different matter. Whether your appeal is successful or not, you will have various costs to bear if you wish to go beyond the District Court level.
Because the answer to your question will be so specifically based on the individual facts of your case, you should obtain a copy of the Decree and have a consultation as to the potential merits (and costs) of an appeal.
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Family Law Attorney
I agree - your question is very broad and, frankly, a little emotionally charged (the "cleaners" statement, for example). The fact is that an appeal can be taken on mistakes of LAW, not of fact. If you believe that the district court erred in their application of the LAW, then an appeal can be taken. Your chances at appeal? Usually, they are slim unless the question of law you are challenging is new, novel, or unusual, or the trial judge was really out to lunch in his/her ruling. The cost of an appeal is tremendous - the briefs that have to be prepared are very large, and the out-of-pocket costs of transcripts, printing, binding, etc. are huge. Expect to spend many thousands of dollars on a appeal, even if your attorney doesn't take you to the cleaners.
This answer is not to be considered a response to a specific legal issue in a specific jurisdiction - it is to be considered only a general response to a hypothetical scenario posed by the questioner. For specific legal advice, please consult with a licensed attorney in your jurisdiction.