You can agree "with" a strategy if you weren't involved in creating or choosing it but you think it's a good idea. If you were involved in the creative process or the selection of the strategy, then you must have agreed "on" it.

It could work either way, but usually we would say we agree with someone on a strategy / course of action, etc.

If we agree with a strategy, it means we think the (already existing) strategy is good. If we agree on a strategy, it means we are deciding what the strategy should be. (I'm agreeing with David K and Johanthan K's answers.)

They are both good, but have different meanings. If I agree with someone, then my opinion is the same as theirs. If two or more people agree on something, then they’ve made a decision together. The city council agreed on a policy to close the parks at night, but I don’t agree with it. (‘it’ meaning ‘the policy’. I could also replace ‘it’ with ‘them’, with ‘them’ meaning ‘the city council’)