Akane
Where is the best spot to go and take pictures of Mt. Fuji in the spring? If possible, I'd like to see some cherry blossoms too. But I’ve heard that there are always a lot of clouds covering the top of Mt. Fuji in springtime so it's not a very good time to go take pictures unless you're really lucky.
Akane
Hakone and the Five Lakes area are both pretty easy to access. Personally I find there's a lot more to do in Hakone once you've seen the mountain compared to the Five Lakes. On the other hand, the view of Mount Fuji is different. At the Five Lakes, it’s a lot more prominent and also the viewing chance should generally be better as it’s closer to the mountain.
As for areas with flowers and a view of Mt. Fuji too, there are spots on the shore of lake Kawaguchiko or the Chureito Pagoda, and the Fuji Shibazakura Festival would be great, but both start around mid April, so it depends if you can go during that season or not.