Apparently Ueno park is a great spot to view the cherry blossoms at the right time of year, unfortunately I was there in May and just missed the hanami festival; not a cherry blossom in sight.
The park is large and offers pleasant walks with peony gardens and a number of beautiful temples and shrines. It's a popular place for Tokyoites to walk and picnic and was very busy when I was there.
You'll find lots of stalls selling octopus balls, green-tea ice-cream and similar delicacies. You an have a cheap and delicious lunch just by buying a few things from these stalls, then sit down on the grass and enjoy it.
There's a concert area beside Shinobazu pond where for a small fee you might get to see some local pop groups play. There are also stalls around the pond selling antiques. I bought some second-hand kimonos as presents.
The park contains the Tokyo National Museum which has fantastic exhibitions of Japanese art, ceramics, textiles and generally beautiful stuff. There's also an archaeology exhibition but I didn't make it that far.
There are some other museums here too, it took me the best part of a day to see the National Museum and another to wander around the park. I didn't even make it to any of the other museums. Plan on spending some time here!