'In', 'inside', and 'into' all have similar meanings with subtle differences.
To start, 'in' and 'into' are both prepositions. This means that they are used with a noun, pronoun, or noun phrase to show direction, location, or time, or to introduce an object
'Inside' on the other hand is itself a noun, in this case, a specific location.
Lets look at the three example statements and break down their possible meanings.
The frog jumped in the well.
This statement is somewhat ambiguous as to the intended meaning. Context might supply necessary details to determine the meaning but without any, the reader would be unsure of the intended meaning. The statement could mean that the frog was outside the well and then jumped into the well. It could also mean that the frog was in the well, and jumped (remaining in the well).
The frog jumped into the well.
This statement's meaning is clear: the frog was outside the well, jumped, and landed within the well. From the previous example, if this was the intended meaning and there is no other context available, the writer/speaker should have used this statement instead.
The frog jumped inside the well.
This statement is somewhat ambiguous. Typically, one would read this statement to mean that the frog is inside the well, it jumped, but didn't leave the well. That said, the statement is sometimes used to imply that the frog jumped into the well; however, this formulation is improper. You may see it used, but you should avoid doing it yourself.