8 days are more than enough for this purpose. OK let me break this post in pieces
Vatican City
is inside of Rome. So visiting Vatican requires only half day if you plan to visit only Saint Peter's basilica from inside out and a round trip of Vatican. But if you plan to visit Vatican Museum, Sistine Chapel, Vatican Library and Gardens etc then it might take a little longer. I don't think it will take more than a day's visit to see the whole of Vatican if you plan to visit everything.
One good hint is when you reach St Peter's Basilica they will ask you if you want to go on top by stairs or lift. I made a mistake by opting to go by stairs (as I thought I will get to see those Spiral Stairs located inside of Vatican Museum and not St Peter's Basilica) and to my bad luck (or good luck as it was a memorable moment), I had to walk about 850 stairs + 350 to reach the top of the dome by foot (stairs are estimated as i remember correctly they were above 1100 stairs altogether. You can google for exact figure). So make sure you opt for the lift option for 5 Euros to save your time and energy. You can also opt to not visit the top of St Peter's Basilica and stay on the ground where almost everything is located but it is nice to go on top.
This is how you see Vatican/Rome from top of St Peter's Basilica...
Rome
OK Rome has tons of attractions but the good thing is, it is still smaller city comparing to your expectations. There are only two underground trains and the route is quite simple. It is not like 12 underground routes in London or 16 in Paris. Most of the attractions are located in Line A with one or two exceptions like Colosseum being found on Line B. Good thing is the route is simple so the chances are very little that you will be lost.
I can give you the full list of places you should visit in the next post if you ask... Like categorise them in must watch to recommended to optional if you want. The whole of Rome can be watched in one day however when you start entering Colosseum and other attractions, it requires couple of days. In my opinion you will be satisfied after 3 days maximum
Venice
Beautiful romantic city but you will enjoy more if you have a good partner with you. Good companionship is most important when you are travelling around. Not many attractions but the area near Grand Canal is stunningly beautiful.
If you have 8-10 days you can also add in one more city if you want.