We visited Monday evening, last entry was at 8:30pm and we got there just before the cut off and managed to buy tickets from the desk inside. Note that their website currently says they're open until 8pm and last entry is at 6:30pm, so I'm not sure if this has changed since our visit 4 days ago, or if the site is in need of an update.
We were offered to leave our coats before setting off (free to do so, I believe). We then had to walk up a few flights of stairs as you start from the top. I was reading the signs on the walls but just before we reached the top, one of the staff members called down to us and started talking to us so I assumed we were being hurried along (although I could be totally wrong and maybe she was just being very friendly!)
We had just started to look at the Playbills on the wall when we were moved to the next room to watch a short video. After that, we were told the rest of the museum would be self-guided and we were moved to the next room.
I was aware of the time on my way around and even though we over took some other patrons, I was still aware that I might not finish looking around everything by the time the museum closed at 10 (and I wasn't sure if it'd be a case of you all get kicked out at 10, or you delay the staff getting home on time - neither is ideal!) so we did speed around it. Their website says most people spend 60-120 minutes at the museum. I believe we were there about 80 minutes and definitely could have done with longer to read more of the signage and information provided.
There are photos of older shows, artifcats and costumes from older and more modern shows (all in chronological order), small model sets, and larger sets you can take photos in.
It felt like a lot of it had been deisgned to be "Instagrammable" and at first I felt like I shouldn't really be taking photos, but we checked with a friendly staff member who was totally fine with us taking photos, and offered to take one of us both on one of the sets.
As a relatively new fan of musicals, I had no knowledge of many of the shows represented sadly, so whilst it was interesting, it didn't mean as much to me to see some of the sets or costumes as it did to see some of the newer things. I can imagine that someone with a broader range of knowledge or experience of Broadway / musicals would get far more out of this museum than I did - but I still enjoyed it (possibly more so because it was somewhat interactive / fun to take photos of).
We didn't realise that each section would have a staff member who was knowledgable and seemed to encourage conversation. As Brits who normally keep ourselves to ourselves at museums etc, this did take us off guard and at first we felt it took away from the "self-guided" aspect of the museum and made us feel a bit uncomfortable. However most of the staff members were so friendly and full of interesting facts, that it actually really added to it the visit, once we knew that's how it worked.
Some staff didn't talk to us at all, and it kind of felt like we were a bit of an annoyance to be in their section but others were very friendly and it felt like we could have spoken to a couple of them for a lot longer than we did, as they were so friendly and knowledgable - including a lovely security lady who stopped and took a photo for us and chatted with us.
Note that I'm not sure it'd be possible to experience this museum if you're in a wheelchair or on crutches as there are many stairs, and I didn't see a lift anywhere (and if you don't go on the stairs, you miss some of the snippets of information).
Overall a very enjoyable museum experience, but just a little more expensive than I think it "should" have been. I assume the high ticket prices are to pay for all of the staff (when we went it felt like staff:customers were about 3:1 between the security, museum staff and shop staff)