Van Dijk, who arrived at Anfield from Southampton in January, insists their faultless start to the campaign is just a small step on a journey the club hopes will end with a first trophy since the 2012 League Cup.
"We want to compete for every competition like we did last year, and we wanted to start well this time," the Dutch defender told British media.
"We've had a good start to the season, nothing has been won yet, we need to show it every week, every three days now.
"Everyone is excited, everyone is feeling good, everyone wanted to be part of this. And you know, we know the season is very long, we have four competitions to play in, and we want to win everything."
Despite showing improvement since the arrival of Juergen Klopp in 2015, Liverpool have lost three finals since the German took over at the club.
"You need to have ambitions, and with the size of this club, with the people around here and the history of the club," Van Dijk added.
"The season is just started, we have started well. We don't need to take anything for granted, we have to keep working hard and stay humble."
Liverpool host van Dijk's former club Southampton in the league on Saturday, before facing Chelsea in a blockbuster League Cup third round clash next Wednesday.
(Reporting by Hardik Vyas in Bengaluru, editing by Nick Mulvenney)
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