Badajoz, the capital of the Spanish province of Badajoz in the autonomous community of Extremadura, is situated close to the Portuguese border, on the left bank of the river Guadiana, and the Madrid-Lisbon railway. The population in 2007 was 145,257.
There is a service station to the west of the city, just before the Portuguese border. This is a great place to get a ride practically anywhere (5 minute wait for Lisbon in 2016, but many cars headed in other directions). You can pretty much walk there from the city centre, but you can also try hitching towards "University Campus"/"Avenida de Elvas", the latter of which is full of roundabouts and lots of slow moving cars. Pro tip: do not go the service station across the border, inside Portugal. Even though it might be slightly closer to your destination, gas prices are more expensive and way more people stop on the Spain side (or even drive over from Portugal) before heading to Lisbon.
Take bus line #1 heading San Roque and ride all the way to the last stop, from there walk about 500 meters along the road until you find a petrol station next to which there is a restaurant called Venta don José. From there you should get easy rides onto the highway E-90.
Take bus line #5 and get out at the last stop of Avenida Juan Sebastián Elcano. Hitchhike at the traffic light where all vehicles have to stop in order to turn right towards Sevilla.