Thinking about moving

So I (28F) am thinking about moving in with my boyfriend and I'm wondering what to expect.
It's my first time moving since arriving in Japan a little over two years ago and I've heard I'll have to change my residency with my ward office and I'm sure I'll have to update my zairyo card and my number card, but is there anything else I should know?
Will I be in trouble because my name isn't on the lease?
Can I expect a new residence tax next year or will it be after two years like when I first moved to Japan?
For reference I'll be moving from Yokohama to Tokyo and I'm giving myself about 6-7 months to get moved and find a new job first.
Thanks!

by a_iry_0