Start with the European hub, not the Bordeaux airport code
Bordeaux–Mérignac Airport, code BOD, has no nonstop service from the United States. A flight from the States therefore needs a connection in Europe. Paris Charles de Gaulle, code CDG, is a familiar option, and Amsterdam, code AMS, is another.
I would start by looking at the complete trip sold as a through booking. The airline takes responsibility for its connecting flight if the inbound arrival runs late, including rebooking you when a protected connection is missed. A separate reservation leaves the traveler hoping the pieces behave.
The temptation is to find the attractive transatlantic ticket, then bolt on a flight to Bordeaux afterward. It might work perfectly. It also means reading separate booking conditions while other travelers are looking at the departure board.
Paris: the plane and the TGV do different jobs
At CDG, a connecting flight can take you straight to BOD. If it is included with the transatlantic trip, I would be comfortable letting the airline handle that handoff.
Paris also gives you a rail option. Some direct TGV trains leave from the airport’s TGV station and reach Bordeaux Saint-Jean without a detour through central Paris. I like the simplicity of leaving the aircraft, completing the airport formalities and heading for the railway station instead of crossing the capital with a suitcase.
But a separately purchased train ticket does not inherit the airline’s promise to get you onto the next service. If the flight comes in late, your rail booking is still your rail booking. Leave a generous gap or choose a flexible ticket; do not treat the train departure as a protected flight connection wearing different clothes.
There are also TGV departures from Paris Montparnasse. Reaching Montparnasse from CDG involves taking the RER and metro across the city with your bags. The airport train removes that crossing. Montparnasse offers more choices of train, but the transfer through Paris belongs in the comparison.

From Paris Orly, there is no flight to Bordeaux. Cross that airport off the flight-connection plan.
At BOD, the terminal name matters before the travel day
Bordeaux Airport uses Hall A, Hall B and the Billi Terminal. The airline determines which building handles your flight. I would read the terminal shown on the booking rather than assume the airport is a continuous indoor corridor.

Work between Halls A and B affects how passengers move between the buildings. Follow the airport signs and allow time if you need to switch halls. A terminal change is not difficult, but a bag does not become lighter because a building plan is being improved.
The terminals also close to the public overnight. Anyone planning to sleep at BOD before an early departure needs a different plan. I would arrange a room instead of treating the arrivals area as a free hotel with security screening.
Use line F for the center and the shuttle for Saint-Jean
Tram line F serves the airport and runs through Bordeaux’s center to Saint-Jean station without a change.

On the way back, read the destination displayed on the tram before boarding. The line also serves stops away from the airport, so the letter by itself is not enough to identify the airport-bound service. I would rather inspect a sign in town than untangle the mistake with luggage later.
If my room is around the riverfront, I would take the tram through the center and finish the hotel journey from there.
The 30’Direct shuttle is a more focused ride between the airport and Saint-Jean. It does not call at intermediate stops. I would favor it when my destination is the railway station rather than a hotel in central Bordeaux.
Taxis wait at the terminals too. They make sense for someone whose actual destination does not fit the tram or station shuttle, though I would decide that from the address instead of defaulting to the taxi queue.
Arrive on a TGV from Paris and you step into Saint-Jean already. I would head for the hotel rather than search for an airport transfer that does not exist.
The passport check may happen before the final leg
US citizens can visit France as tourists without a visa for short stays under the Schengen rules. The Entry/Exit System is operating at the external border: non-EU visitors provide fingerprints and a facial image, with digital records replacing the passport stamp.
For a connection through Paris or Amsterdam, that border procedure happens at the hub before you continue to Bordeaux. A short connection that looks comfortable on a map still needs room for the person at immigration to finish the job. EES is part of border control, not a separate application to buy before departure.
ETIAS is not operating yet, and no applications are being accepted. Ignore anyone offering to sell it for this trip.