Hibiscus syriacus aka Rose of Sharon.

Those are seed pods, not buds. Should bloom once, probably in July where you are. It would be easier to prune for shape as needed in the spring, when it has no foliage. No other care should be required. More trimming = more suckers. Less trimming = a more tree-like form.

We just moved in and I have no idea how to prepare them for the Chicago winter.

==>> ignore them!!!

or get rid of it IF it is considered invasive in your area ...

ken

ps: anything of that size ... most likely.. needs nothing from you.. ever .. as far as care ... a little selective pruning now or then ... to perfect its shape .... is not included in that ...

Thanks a lot, everyone. Much appreciated!