Hi Nolan, welcome to this Forum.
Yeah, there are actually loads of things to do in Myanmar and in these three cities.
Starting with Yangon, of course you already noted the Shwedagon Pagoda on your schedule, which is an absolute must do!!!
Then Chauktagyi pagoda is interesting as well and not very far from Shwedagon. The pagoda itself is nothing impressive, just a big hangar. But it covers a huge reclining Buddha.
The second must do in Yangon is probably its very famous Bogyoke Aung San market, so many stalls in this market, where you will find loads of things to buy. From the handmade items like statues, garments, lacquerware, paintings to the everyday life items like longyi, sandals or shampoo, you can find almost anything there. Be aware that it is closed on Mondays, and closes its doors at 5:00PM
Take a walk in downtown area, where you have the most colonial buildings (around Mahabandoola garden street, Pansodan street, south blocks) and wander around near the river. If you are interested to see the colonial heritage left to the city.
While doing this tour, you can stop at Sule Pagoda which is the central point of the downtown area and an interesting temple to visit as well.
West of downtown, around 19th, 20th street you can explore Chinatown as well and it's street market. Day and night they have markets selling foods of all kinds. If you arrive there at night, stop in one of the barbecued skewers restaurant in 19th street, to enjoy a very local atmosphere, since many people come here at night to have dinner and enjoy some beer with friends.
What else to see in Yangon… Mmmmm, it can very nice to talk a walk in Kandawgyi park around Kandawgyi lake as well. It is an artificial lake, but it is still a very nice park and if you go to the SouthWestern end of the lake (near Karaweik Palace floating restaurant) you can see the Burmese youth wandering around to have a few beers, or listen to some live music on the stage placed there. When the night comes, you can also see the Shwedagon Pagoda from far, illuminated and reflecting in the waters of Kandawgyi, beautiful!
Botataung where the first relic of Buddha in Myanmar is exposed is also worth going, you'll see many people willing to see this hair relic, even if you can't see much actually. The area there is nice, since you have a jetty not far as well (the Botataung Jetty). The pagoda itself is nothing sensational I think, but it's more the symbolic value it has that is worth of interest.
There are certainly other things worth of interest, but I think you already have quite a lot to do with that program already .
You could go to Inya Lake for example if you have time.
Then for Mandalay, as it was mentioned on the forum already, the most interesting things to do are not in Mandalay itself but in the area around Mandalay.
First and foremost, is U Bein bridge. It's the longest teak bridge in the world I think and it is beautiful as you see many people crossing the Lake using this bridge on the first and last lights of the day, a must see.
From there you could go to Inwa, which is a former capital of the Burmese kingdom, and not far from Amarapura where you have U Bein's bridge.
Sagaing and it's pagodas on the other side of the Irrawaddy river, Mingun and its famous bell, a cruise along the river from Sagaing to Mingun. Many things to do around Mandalay.
In the city itself, what is worth to do is the Mahamuni pagoda with its deformed Buddha (deformed buy the many kilograms of golden leaves that were stuck to it) and the Mandalay Hill, nice to climb to have a beautiful view for the sunset.
Last but not least, Bagan is surely one of the highlights of any trips to Myanmar.
As you said there are obviously the pagodas to visit and so many of them, be sure to visit the most famous ones like Ananda, Dammayangyi Pahto, Sulamani, Thatbinyu or Shwesandaw pagoda. Pyathada Paya is also a good one to visit for the view, you have a large terrace on the top from which you can see all over Bagan, I haven't tried the Bagan Observatory Tower though, the view might be ok from there as well, but I think you have to pay to go up.
In Bagan, you can also have a short cruise on the river for sunset. You can reach the small village of Myinkaba by road, where you will find tens of lacquer ware shops and factories, the traditional handicraft from Bagan (must buy as a souvenir).
In Nyaung U (the city where you have bus station and airport) you have the beautiful Shwezigon Pagoda, which is more recent than the pagodas you will have in old Bagan, but still magnificent. Htilomino and Upali Thein temples will be on your way to Nyaung U, so also worth a stop.
Around Bagan there are some sites/attractions as well like Mount Popa and its Nats, and you can go to a Palm tree plantation/palm sugar factory, also. These will require a car though.
I certainly haven't explained all, but you surely can have a good idea already about what there is to do in these three places.
If you need to read more about it, here are a few links from the forum on Bagan, Mandalay and Yangon. It sure can give you some more information:
Best Things to do in Bagan
Sight seeing in Mandalay ; trip in Mandalay, Sagaing and Mingun
Two days in Yangon
Have a great trip,