Chinapatern,
There are many possibilities for the journey you plan to Pisa and Rome. KLM, AF, SN, LH to name a few. However my tip would be Swiss, I've flown many times BHX-FCO with Swiss and it's an excellent product. Free baggage, choose your own seats, pre-check-in, free bar and food service on board and ZRH is a delight to transfer through. I think you'll find the fare very competitive with the others, and especially so if you pay the loco add-ons. You will also have a choice of three departure times daily.
Swiss is one of the most pleasant ways to fly in economy and the short haul product is very good, and if anything goes wrong they look after you and take all the hassle out of the equation, they have good fares on their own site. Give it a go.