Appeals are fairly complex and differ from district court proceedings. There is no new evidence in an appeal, which means that witnesses are not called and exhibits are not entered. A party often gets a new attorney on appeal, and is sometimes required to do so. If you are the losing party and appeal a judgment, your new attorney orders the transcript from the prior case. The attorney from the trial is typically the one who files a notice of appeal in the district court. There is a strict timeline for filing an appeal. A party commonly has only 15-30 days to file a notice of appeal.
Appellate courts ordinarily review the thought-process of the judge or jury's district court decision. If the judge made an error of law or the jury came to a conclusion that no reasonable jury could conclude, and you are the losing party, you can appeal (so long as the error is preserved). Preservation of error requires an objection in district court or other notification that the trial attorney disagrees with a decision on a particular issue. In some cases, appeals are standard procedure, and the losing party does not need to preserve the error.
Sometimes, you can appeal a non-dispositive order, which is one that does not end a case in its entirety. Not all non-dispositive orders permit an appeal. For instance, if a motion for class certification is denied, the aggrieved party can immediately appeal, and must do so within a limited time. If an order is entered that partially decides a case, the aggrieved party need not appeal until after the entire case is finally decided at the district court level.
As you can see, there are many considerations in an appeal, and this is simply when it comes to initiating an appeal. After an appeal is initiated and the transcript has been ordered, the appeals court will set a timeline for writing legal briefs and oral argument (if requested and/or necessary). State and federal jurisdictions differ on the appeals process. In some states, filing a notice of appeal sends the case to the state's supreme court, who might resolve an appeal or send it to its court of appeals, if that state has a court of appeals. Some smaller states do not have courts of appeals.
Under federal law, the procedure is different. Filing a notice of appeal in district court sends the appeal to the regional circuit court of appeals. Then, depending on the disposition of the case at the circuit court level, a party can petition for certiorari to the U.S. Supreme Court. "Cert," as it is commonly called, is infrequently granted.