'Since' only refers to a time period starting in the past and leading up to the present. If you want to express the same idea for a different time period - in this case, the future - you can use 'from': "We'll continue our lessons starting from next week".

"We will continue our lessons starting next week" is correct. "Since" would be used for a time in the past e.g: "I have been taking lessons since last week"

We will continue our lessons next week. 'Since' refers to a specific time in the past. I have been taking lessons since last Saturday. I have been eating grapes for lunch since November last year. Etc Hope that helps!!!