Recently we plugged our toilet. We unplugged it using a plunger. However, subsequently the bowl was not filling completely and wasn't effective in flushing deposits. So I changed the fill valve and now the bowl fills. However, the flushing action is weak; it doesn't flush with the same power as it did before.

I suspect there might be a partial blockage in the pipe somewhere that reduces the suction used to create a powerful flush. How can I solve this problem? Would a drain snake be the answer? If so, I have had a hard time finding one that is flexible enough to accommodate the curves experienced in a toilet - any suggestions?

I suspect there might be a partial blockage in the pipe

I would agree.
To snake the line you have to remove the toilet. With the toilet removed a drain snake or even a sewer snake will make the turns.