How would air travel between Texas and the U.S. work?

The same way air travel works between any two friendly countries, which is to say easily and constantly. Planes fly between countries every minute of every day. The whole world runs international air service on a settled framework of treaties and agreements, and an independent Texas would plug straight into it. You would still board a flight from Dallas or Houston to anywhere in the United States; what changes is the category of the agreement underneath the flight, not the flight.

International air travel already runs on a global rulebook

There is nothing improvised about flying between countries. Since 1944, international civil aviation has operated under the Chicago Convention, which created the body now known as ICAO and the shared rules that let aircraft cross borders safely. 193 countries are parties to it. An independent Texas would join that framework as a matter of routine administrative work, the same way every new nation does, and Texas aircraft, airports, and pilots would operate under the same internationally recognized standards as everyone else. The system that moves people between countries is mature, global, and built precisely so a flight from one nation to another is ordinary.

Service between Texas and the US runs on an air-services agreement

Commercial flights between any two countries operate under a bilateral air-services agreement, and the open, liberalized version is called an Open Skies agreement, which removes government limits on routes, how many flights, how many seats, and what fares airlines can offer. The United States already maintains Open Skies agreements with well over 125 partners around the world, because more flying is good for the US economy. Texas, with two of the busiest airports on the continent and an economy deeply tied to the United States, would be negotiating from real strength for exactly that kind of open arrangement. The result Texans would actually notice: the same airlines flying the same routes between Texas and the United States.

The deep ties are the reason to keep flying easy, not a barrier

Some will argue that because Texas and the United States are so connected by air, independence would tangle it up. It cuts the other way. Precisely because so many people fly between Texas and the rest of the country for business, family, and tourism, both governments have an overwhelming interest in an open-skies arrangement that keeps those flights full and frequent. American carriers want access to Texas markets as much as Texas carriers want access to American ones. Choking air service would cost both economies. Mutual interest points hard toward keeping the skies open.

The passport-at-the-gate question is already answered

Whether you personally need a passport to fly to the United States is its own question, and the live site answers it in full: many Texans keep US citizenship, contiguous neighbors keep travel easy, and the federal Visa Waiver Program already lets citizens of dozens of countries fly to the US without a visa. We will not restate all of that here. The point for air travel specifically is that the routes, the carriers, and the airports keep operating; the document you carry to the gate is covered in the dedicated passport answer.

The bottom line

Air travel between Texas and the United States keeps running, on the same global aviation framework every country uses and an open-skies-style agreement both sides have every reason to want. Texas joins ICAO like any nation, the same airlines fly the same routes, and independence changes the agreement under the flight, not the flight itself.