continue by doing something
to keep going by starting to do something else or the next step. You are doing very well in this piano lesson. Please continue by playing the other sonata. After the interruption, Wally continued by explaining his position on the trade negotiations.
continue with something
to keep doing whatever was being done before. Oh, please continue with your discussion. Do you mind if I continue with my knitting as we talk?
Could we continue this later?
and Can we continue this later?Could we go on with this conversation at a later time? As Mary and John were discussing something private, Bob entered the room. "Could we continue this later?" whispered John. "Yes, of course," answered Mary.