No, for the most part they didn't, the vast majority of candidates ran so-called "front porch campaigns" that were conducted from their personal homes. They would occasionally hold rallies for visiting delegations there, but not travel much. This was both because it was just logistically tough to criss-cross the country before train networks became better developed, but also because it was seen as undignified for candidates to campaign personally and ask for votes rather than just wait to be asked to lead the nation.

However, parties heavily used surrogates to give speeches, both local leaders and national politicians with good oratory skills who did travel giving speeches in support of their candidate.

Campaigns were also conducted through the media, which for the most part was much more explicitly partisan then than it is today--many newspapers were explicitly linked to a political party, right down to the name: Just a brief look at the newspapers archived by Google (https://news.google.com/newspapers) returns 40 newspapers with "Democrat" in the name and 29 with "Republican"-- while these archives are more complete for the late 19th century than for the antebellum era, and obviously the Republican Party didn't form until just briefly before the Civil War, it still was much the same in that era.

Even the New York Times was an explicitly Republican newspaper for most of the 19th century.

Actual campaigning (which in many cases amounted to handing out civil service jobs and vote-buying) was left to local and state party chapters, a famous example of which was the Tammany Hall Society running the Manhattan Democratic Party.

Campaign songs, which were usually giving new lyrics to already famous melodies, were also a big part of getting the candidates and their messages out to normal voters. The most successful of these might have been "Tippecanoe and Tyler Too", used by William Henry Harrison in his campaign against President Martin Van Buren in 1840.