I am planning to spend 2 days in Milan and want to visit Leonardo da Vinci’s Last Supper, the Milan Duomo, including rooftop tour, the Scala Opera House, Sforza Castle and the Galleria. Is it easy to get about from place to place via metro and secure entry to these venues on arrival (except for Last Supper-advance ticket required) or is it best to pay for a tour?
In May 2024 I managed to visit all of the sites you mentioned using public transit. But I walked from the Duomo to La Scala; it's 10 minutes maximum on foot. Also the Galleria V. Emmanuele is only a minute walk from the Duomo.
If you want to go up on the roof of the Duomo, I think you should consider pre-booking a timed entry and plan to make that your first stop at the Duomo complex. That way you'll know when you can get started and will avoid the security line for the church itself--though the latter wasn't burdensome in August. I had no trouble just buying a ticket for the rest of the Duomo sights at the ticket office on the square (not right at the church).
There are a lot of separate museums at the Sforza Castle, most if not all covered by the same ticket. You probably won't want to see all of them since you have limited time in the city and have several other things you want to do. I recommend taking a good look at the website ahead of time and plotting your sightseeing strategy. The Castle can take a lot of time if you try to see everything.
I used a good bit of public transportation during my five days in Milan. It wasn't difficult. The CityMapper app was helpful in getting around. It shows you walking time as well as transit directions, so you can decide whether you'd rather just walk.