have you looked at train schedules? Venice to Cinque Terre is at least 5 hours just on the train and at least 1 train change (depends on which CT village you choose). It's about the same from the CT to Rome. to Rome from the CT, you have to take a local train to La Spezia. There are 2 fast trains a day from La Spezia to Rome (travel time 3 hours). One leaves at 7 am, the other at 8 pm. Remember that before you board the train, you have to pack, check out, get to the train station and find your train. Then when you get off the train, you have to take your luggage to your hotel and check in. So the early morning train probably means getting up well before 6 am and only rid of your luggage at 11 am. Taking a later train means at least another hour travel time and getting into Rome hours later. This is the problem when you are planning to visit locations that are farther from the major cities and without the convenient rail connections those cities enjoy.
How many locations you end up with is a personal choice. Most, though far from all, of the folks here prefer a slower pace. As for fishing, it's not something I've ever been interested in doing. However, I do love to read books. And while I am sure I'd enjoy reading a book more on the shore of Lake Como rather than in my living room, the minor enhancement would not justify spending my time in Italy in an activity I can darn well do without long flights and expensive lodgings. Using Google, I found this website to give you an idea of possibilities and prices.
And speaking of lodging, your trip is only 2 months away and it's high season in Italy, so start looking for hotels soon! I don't understand your comment that staying "outside of hotels" would allow you to "experience the culture." Do you mean staying with families? If you rent an apartment, consider that you may have very limited arrival and departure hours and you will not have 24/7 staff to answer questions, give directions, and offer suggestions. Other than that, there's little difference between a hotel and apartment experience.