Unfortunately very few. The Visigoths as a people ceased to exist after the Muslim invasions of 711. It was only in the late 15th century that the Muslims were expelled. In that nearly 700 year period, Muslim culture dominated and displaced most Visigoth practices with their own. Prior to the invasions, the Visigoths maintained some aspects of old Roman infrastructure and class systems, much like the Ostrogoths in Italy under Theodoric, which worked rather well. Roman Coloni farming estates, for example, continued to survive under Visigoth rule. However, because they sought to preserve the roman infrastructure that worked so well, they didn't make many radical chances to the people or land themselves. And again, any changes they did make were overwritten by 700 years of Muslim rule.

Saint Isidore of Seville was a Spanish scholar who produced perhaps some of the most influential works to come out of Visigoth Spain. His "Etymology" was a collection if words and their supposed origins. Most of what he says is not at all accurate, yet it remained popular with Western European monasteries and referenced excerpts from classical literature which would otherwise have been lost. He also helped to compile the "collecto hispania", an influential collection of Spanish common law.

Toledo was also made the capital city of the Visigoths and was noted for being a place were christians, Muslims, and Jews co-existed after the Muslim invasion.

Those are the only examples I can think of, there are others I'm sure but the Visigoths in the end had very little influence on Spain in the long run.

Source: Western Europe in the Middle Ages, 300-1475 by Brian Tierney

Tangentially, are there many Visigothic words that have survived into the modern Spanish?

A somewhat short but to the point answer to this question is: Almost nothing.

Remember, Spain was heavily populated already when the Visigoths invaded. Culturally they weren't some backwater. They had assimilated Latin culture, had centers of learning and even contributed a few Emperors to the Roman throne.

The Visigoths arrived and were then largely kicked out within a span of around 200 years.

You also have to keep in mind that Visigoths were of the Arian sect of Christianity while their subjects were largely Catholic. I remember reading that the Visigothic code of law prohibited marriage between the two, meaning people had to convert or remain unmarried. This impeded the full assimilation of the Visigothic ruler class by the dominated Iberian natives.

Some Spanish historians even propose that the lack of unity fed the armies of invading Muslims with fresh recruits who welcomed aid in deposing their heretical overlords.

Sources: The Age of Faith by Will Durant Visigothic Spain 409 - 711 by Roger Collins. The Civilization of the Middle Ages by Norman Cantor