My question for the following two sentences is, which is correct or better.
1) I would like to carry on the program further by doing ...
2) I would like to carry the program further by doing ...
First, carry on means to continue doing what you are doing, and it works like this,
Ex: Carry on doing something.
Ex: Carry on with something.
1a) I would like to carry on with the program by doing....
1b) I would like to carry on, further, with the program by doing...
Second, carry...further means to start something new.
All the best.