Genoa gave me an impression of rather unattractive industrial city/port as i drove through - probably there are more interesting and pleasant areas there but you need to be selective. If CT is of interest i'd stay in Rapallo and explore the nearby area. Rome is a large city CT has more of nature appeal - personally i'd pick CT area to supplement florence as you get enough history/art/urban feel etc etc there but there are pros and cons to both. April should be less touristy especially in CT but the weather factor can be significant - rainy/windy CT is not gonna be pleasant or photogenic.