I don't think there is one right way of doing this - you just have to find a compromise that works for you both.
I would never expect my husband to finance anything related to my child (from a previous marriage). He buys her stuff or pays for a meal or day out or whatever for her occasionally but that's as a gift - I would never expect it. So on our upcoming holiday, tho we won't work it out totally strictly, the idea is that I will pay roughly two thirds, if not more. I earn more than him at the moment, so to expect him to pay more than his share would seem pretty awful to be honest.
However, in a previous relationship I did contribute financially towards my step children in terms of holidays etc. Although their dad was pretty rich, I chose to do that because I felt their father was ungenerous towards them to be honest, and I was close to them (still am close to some of them).
My dd's dad (this is a different past relationship of mine, not the father of my former step kids) is now with a woman who has several children (not his) and according to dd he pretty much pays for all the group holidays etc. He is very well off compared to her so that doesn't surprise me - he is probably happy to do it in order for them all to enjoy the kind of holiday he likes. (don't get me wrong - I'm not saying he does it selfishly - he is generous).
None of these situations is wrong or right. They are just what worked at the time. Nobody can tell you what you should do here. It totally depends on the dynamics of your relationship.
One suggestion I would make is, do you need/want to do all the activities together as a five? Or would it make sense (both financial and emotional) for your husband to have a couple of afternoons, say, where he does activities that interest his kids, but not so much your joint kid, with just them? Nice for them to have quality time with their dad, and it saves money. I will certainly have a day or two on holiday that I spend just with my daughter - for example we love the fairs and arcades, and my husband isn't bothered at all, so why drag him round/waste money on his entrance fee?