Mostly as treasures and valuable Items. They did not have archaeologists, in the sense of people who dug and analysed their findings for science. There was such a thing as coordinated excavations but again only to find treasures or such things. For example in Corinths which was destroyed by the romans and later built up as a colony (46 B.C). When the city was built up the colonists found pieces of pottery in the old graveyard and sold them for a bunch of money to Rome (read Strabon for more on that and corinthian history) and called them necrocorinthiae (necro=dead corinthiae= stuff from Corinths). Source: I study archaeology and had to give a presentation on the topic about a month ago. Also excuse my bad english it is not my first language.

There's a great book on the history of paleontology called The First Fossil Hunters. I don't want to mislead you because paleontology and archaeology are different, but the Romans did understand that there were ancient forms of life that were no longer around and incorporated them into their mythology as giants, heroes, etc.

Archaeology, as it exists today as a social science that relies upon the scientific method to inquire of the past, was actually begun by Thomas Jefferson.

In 1780, a French diplomat submitted a list of questions concerning Virginia's native populations. Thomas Jefferson was tasked with providing an answer. One of the questions dealt with burial habits. Instead of repeating what was commonly thought of the time, Jefferson decided to excavate a burial mound on his family's land at Monticello. While digging, Jefferson developed the concept of stratigraphy, which is still the modus operandi that we use in archaeology today. We he realized is that every time the earth in which he dug changed, it represented a subsequent change in the geological narrative. In other words, if the dirt changed from a red, clay-like substance to a layer of ash, Jefferson realized that this represented a period in which a fire occurred and left the ash deposit.

So in the modern sense of the term archaeologist, the field did not exist in Roman times.

If you are interested in the history of archaeology, I would suggest checking out the Monticello website, which has an extensive section on Jefferson's excavation. It does gloss over certain facts, however, such as the native population's mound that Jefferson dug likely still existed. In his publication of the dig, he relates a story of seeing members of the native group visiting the site as a boy. This gets into all sorts of ethical questions.

In terms of digging up old stuff, the epodes of the poet Horace seem to suggest that, at least in his time (ca. 30 BC), so-called "witches" would dig up bones and other objects from what they perceived to be ancient graves in the Esquiline Necropolis (which dates back to the 7th century BC) for use in their magic. For instance, see the character of Canidia in his Fifth Epodes. Although it goes without saying that Horace's claims can't be taken at face value, especially since he does all he can to make the witches look ridiculous and tries to debunk the idea of witchcraft in general, there certainly are scholars who accept, at least in general terms, the premise that Romans of the late Republic/early empire would sometimes dig up objects and/or bodies from their distant past (for a Roman living in the first century BC, the 7th century is eons ago) but the practice doesn't seem to have been too widespread or acceptable, due to cultural and religious reasons. If you're interested in reading an argument which suggests that Horace's witches and their methods of practicing magic were largely non-fictional, see C. E Manning, "Canidia in the Epodes of Horace," Mnemosyne 23, no. 4 (1970): 393-401. Of course, what I've mentioned is more just an instance of regular old grave-looting rather than archaeology, but in the earlier days of archaeology the distinction between the two was slim anyways.

Although the Romans had no qualms about destroying, burying or radically changing certain buildings and structures when it served their purposes, they also could practice some degree of historical preservation for buildings or locations that were culturally significant or that served Roman identity and Roman values in some way. In the first-century BC, the Romans maintained a sort of open-air museum on the palatine hill (which is where archaeologists have found the earliest indications of settlement). This open-air museum is mentioned in a number of sources and consisted of an ancient hut, the Casa Romuli, which reportedly belonged to Romulus, the mythical founder of Rome. The hut these Romans made so much effort to preserve (it was saved from multiple fires) may have never belonged to Romulus (and Romulus likely never existed), but it's still pertinent to your question to note that yes, there was at times (namely when it served certain ideological purposes as in the case of promoting a specific foundation myth) an interest in preserving and caring for ancient sites/ruins that they recognized as being from the past.