It's only 12 days since the win over Munster, but a large section of the squad returned to training on Wednesday and another group will get started next week.
Speaking at Glasgow's open-top bus parade in George Square, Finn Russell told BBC Scotland: "You don't have too long to savour it.
"It is a quick turnaround and we are back in for the start of pre-season on Monday."
Finn also admitted to being a bit overwhelmed by the welcome in George Square.
"I didn't expect anything like it," he said. "It is amazing, especially as it's during the week.
"It's amazing to have all the fans come down, especially coming down Dumbarton Road,
"These are the moments that will be highlights of your career and you've got to make the most of it.
"Expectations are going to be high after winning it."
The young scrum-half hopes that he and coach Gregor Townsend can continue the PRO12-winning partnership beyond next season.
"I've got another year of my contract and hopefully I'll manage to get something else after that," he said.
"Gregor and the coaching staff are brilliant. I am loving the club.
"He (Gregor Townsend) has been a brilliant head coach the last couple of years and, with me being a young 10, it's been great to have him around.
"Glasgow are going in the right direction and it's a brilliant club to be in.
"The boys are just like one big family and you can see on the pitch how much effort they put in for each other."