What is the best most direct route to take to Rome in mid April? Looking for advise as we are then traveling to Venice.
The answer depends on what exactly you mean by "best" and "direct". Shortest in terms of travel time will be to take a regional train to Ventimiglia (first Italian station after the border), then InterCity to Genoa or Milan and from there high-speed Frecciarossa, Frecciabianca or Italo trains to Rome, with these options you can roughly count on roughly 8,5-9 hours travel time in total. There is also a direct InteCity Ventimiglia to Rome (meaning that you'll have to change train only once), but it is really early in the morning 6:33am, so you'll have to start from Monaco at 5:48am to catch it
In terms of ticket fare (however here will be good to know your exact travel date) option with direct Intercity Ventimiglia-Rome will be the cheapest most of the dates, however via Milan or Genoa it is as well possible to travel relatively cheap in case you are ready to purchase your train tickets (=discounted 'Super Economy' or 'Economy') today via trenitalia website
Rome to Venice is easy and can be rather cheap with already mentioned discounted tickets (start from only 19 euro), again either TrenItalia high-speed trains or private company Italo
p.s. does sequence Monaco-Rome-Venice is a must? since a bit more logic seems to travel Monaco-Venice-Rome, then you won't need to travel Norther Italy-Rome there and back, but only once