Monday, 15 August is a huge holiday in Italy, Ferragosto, so I'd be quick about setting your plan and reserving your hotels for your arrival.

Since you did not book a multi-city airline ticket and must return to Rome at the end, I suggest you think about your first stop as "not Rome," putting all your Rome days together at the end of your trip when you must return to Rome to fly home. This will save one hotel change—hotel changes usually waste at least a half day with checking in/out and traveling.

For example, since you want to visit Venice, upon arrival at Rome airport, take the train (30 mins) to Roma Termini station and then take a train to Venice (4 hours). You would arrive by dinner time. Spend your desired number of days in Venice and then work you way south back to Rome.

Another example: upon arrival at Rome airport, take the train (30 mins) to Roma Termini station and then take a train to Florence (90 mins). From there you could head north to Venice and then south to Rome. Florence is a good place to experience Renaissance art if that interests you.

Or, at Rome airport, rent a car and head to Tuscany for a few days, staying in a small town. Then dump the car and take the train to Venice, and so forth.

Ellen is correct---you should have gotten some counsel before booking the flights. It will cost you almost a day.
I would pick 3 destinations in that time--it seems you have 2 picked already. My third would be rural Tuscany---with a car.
August is hot and 3 cities would be too much for me. Click on my name to find my recent trip report in Tuscany.

Ellen:

I was unaware of the holiday. As far as saving Rome for the end of the trip, I think that sounds like a good idea and hadn't thought of that. I really like the idea of starting in Venice. Thanks for your help.

Bob:

Thanks for your advice. I would ask you another question. Is there a way to do something similar to your Tuscany trip without driving? I am trying to avoid renting a car if possible. Also, would you be able to pick between a hill town in Tuscany and the Amalfi coast?

Thanks

I agree with Bob that for your 10 days (not including the two travel days), no more than three locations would be comfortable.

Logistically, Tuscany is between Venice and Rome, so it would be easier to connect these three.

Connecting the Amalfi area to Venice takes the good part of a day and a series of connections—doable, but be prepared to be traveling for 6-8 hours depending on connections and location on the Amalfi Coast.

To see Tuscany without a car, you could base in Florence and daytrip via bus or train to some Tuscan towns. Or to get a small-town hill-town feel, base in Siena and visit hill towns via bus

Rome will be blazing hot and Venice will be hot as well as humid. I think Ellenm's suggestion of opting for a stay in Siena would be a good one, especially if you can book somewhere with a swimming pool - perhaps this one?
http://www.aiamattonata.it/

Ok, so after a bit more research and your suggestions I have come up with a skeleton itinerary as follows:

Day 1 AUG15 (partial day) Arrive in Rome, ride train to Venice, check in to hotel, dinner.
Day 2 AUG16 Venice - gondala, bridges
Day 3 AUG17 Venice - piazza sam marco
Day 4 AUG18 Leave Venice - Travel to Sienna
Day 5 AUG19 Explore Sienna
Day 6 AUG20 Daytrip to Florence - want to see the Duomo
Day 7 AUG21 Either depart for Rome or another day trip maybe to Pisa or Lucca
Day 8 AUG22 Rome
Day 9 AUG23 Rome
Day 10 AUG24 Rome
Day 11 AUG25 Rome
Day 12 AUG26 Leave Rome

for the Rome part of the trip I would like to focus each day on a geographical area, with some time devoted to popular tourist stuff, some time for off the beaten path, some eating then night life near a hotel. If you could point me to a reference for such things or have suggestions, awesome. Also, what factors do I need to consider when picking what part of Rome to get a hotel?

I agree that Tuscany would be a better choice than the Amalfi Coast because at the time you will be there, everyplace will be crowded but I expect AC most of all and it is a pretty good distance from Venice. Tuscany is best seen with a car unless, as ellenem says, you base in Florence or Siena.

For a honeymoon a small agritourismo with a pool sounds great and with a car you could visit small hill towns around your base. Italy will be hot and crowded in August so a more remote location would be more relaxing.

Whatever you decide, you need to make accommodation reservations ASAP.

If you day trip to Florence do not miss Michelangelo's David at the Accademia. Be sure to make reservations in advance.

Also, is it advisable to get an apartment in rome or a hotel?

What Ellen said---the AC is too far and complicated and will be wall to wall with Italians. Your basic problem is August.

For Tuscany, base in Siena and take day trips by bus to your choice--not the same as a car, but it will work. Or,hire a driver one day to hit the villages.

A worthwhile splurge for your time in Siena would be to book a day with driver/guide Luca of hillsandroads.com. He can drive you to the cutest of the cute/quaintest of the quaint hill towns, or vineyards, or monasteries. You can give him a general idea or be very specific and he will create a wonderful day for you.

The towns of Pisa and Lucca are a bit far from Siena compared to some other hill towns that you might enjoy. Time to do some more research.

To throw into the mix: if you plan to visit Galleria Borghese in Rome, it's worth knowing that you'll need to reserve your ticket ahead of time. It's not possible to buy a ticket on the day of the visit. Here is the website for further info: http://www.galleriaborghese.it/info-en.htm

And as per mamcalice's advice, book ahead for the Accademia (and the Uffizi if you are cramming art in) in Florence otherwise you'll spend far too much time waiting in the ticket queue and not enough seeing the actual art.

Wow, tons of good advice. Combined with some more research here is my current tentative schedule:


Day 1 AUG15 (partial day) Happy Ferragosto! Arrive in Rome, ride train to Venice, check in to hotel, dinner.
Day 2 AUG16 Venice: Focus: Main Sights
Morning: St Marks square and the Doge's Palace
Lunch:
Afternoon: Take the Vaporetto #1, Rialto Bridge
Dinner: Random Place found after wandering
Day 3 AUG17 Venice: Focus Getting lost in Venice
Morning: Glass blowing tour in Murano
Lunch:
Afternoon: Return to St Marks Square and begin wandering aimlessly
Diner: Random
Day 4 AUG18 Leave Venice - Travel to Sienna
Day 5 AUG19 Explore Sienna
Morning: Piazza del Campo
Lunch:
Afternoon: Museo dell'Opera
Dinner:
Day 6 AUG20 Daytrip to Florence:
Morning: ride to Florence, Accademia, Bargello Museum
Lunch: in Market Neighborhood
Afternoon: Uffizi Gallery
Dinner: after walk through Oltrarno
Ride back to Sienna
Day 7 AUG21 Day Trip to Tuscan Hill Villages with driver/guide Luca (thanks ellen, still working on reservation for this though)
Day 8 AUG22 Rome: Focus: Ancient
Morning: Capitoline Museum, Imperial Forums, Colosseum
Lunch: near Colosseum
Afternoon: Roman Forum, Pantheon, Spanish Steps, Trevi Fountain
Dinner: Take in dinner in Piazza Navona
Day 9 AUG23 Rome: Focus Museums
Morning: Vatican City and Museums
Lunch:
Afternoon: Castel Sant'Angelo, Galleria Borghese, wander around Villa Borghese
Dinner: somewhere in the area
Day 10 AUG24 Rome:
Morning: Out of Rome Journey to Ostia Antica
Lunch:
Afternoon: Appian Way and Catacombs
Dinner:
Day 11 AUG25 Rome:
Focus of Today "get lost" wondering around a neighborhood (Trastevere) need ideas on this one
Day 12 AUG26 Leave Rome

Would appreciate any suggestions for eating establishments, for the most part I want to rely on chance to find lunch places.

Sounds like a great honeymoon HtownResider!
For a hotel location in Rome I would recommend the area around Piazza Navona, you can walk or take short cab rides to most things on your itinerary and there are lots of choices for restaurants. Plus there is always something going on in the Piazza that is entertaining and the people watching is good.

I was just there for four days at the end of April and we stayed at the HOtel Raphael which was wonderful. It included a good breakfast, rooms had air conditioning as well as big windows that opened, comfortable beds and helpful staff on the front desk. We had room 202 which overlooked the front of the hotel and was a recently re-modelled room. It was a gorgeous room with a spacious bathroom with separate tub and shower.
My brother and his wife stayed not too far away, closer to the river and Campo di Fiori at the Ponte Sisto and they loved their hotel also.
For a first time visit a hotel v.s. an apartment is what I'd choose.

Near the Piazza Navona we had a great casual dinner one night at Cul de Sac, fantastic selection of wines. Then you can continue down that street to Frigidarium for some delicious gelato.

Trastevere is an interesting neighbourhood to wander in, we stayed there last time. It is like wandering in a labyrinth!
There are some good restaurants there, Le Mani in Pasta being one.

We also had a few days in Venice and stayed at the Locanda Orseolo which I can't recommend highly enough. The staff there are so friendly and helpful you feel like you are part of the family. They would be able to give you good dining recommendations.

Sorry I haven't included the links (not too technically oriented) but you can find all of these places through Google.