I take a different view from the posters above. I would be looking to check raise this flop. By just calling behind we open ourselves to another diamond coming that not only puts us in no mans land on how strong our hand is or if we are up against a flush but it also kills the action against opponents who could easily have a one pair hand here (Ax) without any diamonds. I would be check raising here not only to protect the hand I have (bottom set is vulnerable) but also to get vale from Ax
hands and anyone who wants to come along for implied
odds with a high one diamond hand.
As played going to the turn I think I would just call as you did since raising here into this size bet is only going to get called by those hands that are beating us whereas on the flop we can get worse to still call. The outside exception would be AK but the villain who bet only overcalled pre flop so I would think if they did have AK then they would have likely 3 bet pre. It makes me think now they have Ax or have the flush so I would just call here and take my showdown value with a set. If the river brings in a paired board then I would lead huge for value on river but if it comes in anything else I would look to check call except another diamond where there would be a decision that would need to be made.
River comes in and I would not be worried about 4x from villain for the straight so I would check as you did and look to call off the bet they make. No reason to raise on the river here for the same reason we are not raising on the turn.