You don't mention the specific type of Continental tyre you currently have fitted, but Michelin Cross Climate are an all-season tyre, so unless your Continentals are also all-season you should not mix an all-season or winter tyre with summer tyres. Having different brands of tyre on your car is less of an issue if they are the same type and dimensions, although ideally you should avoid mixing brands on the same axle.

If you are only changing two tyres, you should fit the new tyres on the back. Having more tread and therefore greater grip on the rear prevents oversteer and loss of control in wet conditions.