The 22-year-old joined Southampton in a Scottish-record £12.5m deal in July after two years with Celtic, the highlight of which was his opening goal in last season's memorable 2-1 win over Barcelona.
Celtic have again been drawn against the Spanish giants in this year's Champions League group stage, which also sees Neil Lennon's side face mouth-watering matches with AC Milan and Ajax.
"Obviously Champions League they are good games, but it was time for me also to move on," said Wanyama. "I will be just looking forward to support them.
"I believe one day [Champions League football is achievable at Southampton]. Everything step by step and I believe one day we will be also there."
Liverpool, Arsenal, Everton and Cardiff were all linked with the Kenyan midfielder during the summer, but it was Southampton who fended off their rivals to land his signature despite initial talks breaking down.
"It was my decision after meeting with the chairman and the manager," he added. "I was just happy with the things they were telling me and that's why I chose Southampton.
"The ambition of the club they had and I was just happy to come and join them.
"It has been a little bit easier to get used to the change, because the team-mates here have been really helpful and it has been easy for me to settle in.
"We have been working hard and we had some good results, and also some bad ones, but I think we are still on track and looking forward to doing well.
"I would say first is to just do better than last season and from there we will see what happens. I think that is possible."