All of the relevant courts, including county courts, are part of the state court system. There respective jurisdictions are set forth in a pamphlet available on line prepared by the Texas Supreme Court in its role as the administrative body in charge of the judicial branch of the state of Texas.
The appropriate court depends, in part, upon the kind of relief you are seeking and the nature of the claim, which is not specified in your question, so it is impossible to answer your question definitively.
District courts in Texas are courts of general jurisdiction that have jurisdiction, at least, over all disputes involving more than $500. They also have jurisdiction to grant non-monetary relief such as injunctions. My guess is that most things you would want to sue a school district over are questions for District Courts rather than a county court, but without knowing what you want to sue over, it is hard to know.
There are actually two kinds of county courts in Texas. Constitutional county courts are limited to disputes involving $200 to $10,000. Statutory county courts have jurisdictional limits with a minimum of $500 and a maximum set on a case by case basis by statute up to $200,000 in some statutory county courts. So, you would need to know which county court was involved to know its jurisdictional limits.
There are also justice courts which have jurisdiction over claims of less than $10,000 which no minimum amount in controversy.
There may be other laws such as governmental immunity acts that also limit where you suit can be filed, but that depends upon the nature of the claim. Suits alleging a violation of constitutional rights by a school official could also be brought (and usually would be brought) in the appropriate United States District Court.
The question of which court of a given type you should file your lawsuit in is called "venue" rather than "jurisdiction." In the case of a school district, the appropriate venue would usually be either the county or district in which the event giving rise to the lawsuit occurred (e.g. in the case of a personal injury), or the county or district in which the headquarters of the school district is located. Usually, if there is more than one acceptable venue, the person who is filing the lawsuit gets to choose which one to file a lawsuit in.
Geographic venue is quite complicated in Texas because it has lots of courts with boundaries that don't fit into neat patterns.