Scotland qualified last November, ending a 23-year wait to compete at a major tournament. Their last was the World Cup in France in 1998.
Tierney said: "It's something the country needed for a long time, a good group [of players] who can qualify for a major tournament. But now we have done that we don't want to stop there — we want to make ourselves proud and, most importantly, the country."
Scotland kick off their campaign on Monday, when they face the Czech Republic in Glasgow. After that, they play England at Wembley and Tierney insists it will be an emotional affair, particularly as he could come up against club-mate Bukayo Saka.
"As a Scotland player it doesn't come much bigger," said Tierney. "I could be up against Bukayo, but I'm looking forward to it whoever I'm up against."