First day in Rome

Trastevere is a wonderful place to start from! I really enjoy wandering around there later in the afternoon (after riposo) and into the evening. You can't go wrong with the food - find a crowded place, it'll feel more like a party. It's very easy to cross the Tiber and be in the Campo dei Fiore and Piazza Navona to finish the evening.

I agree about Trastevere. Just hang in the neighborhood where all the crowds are. It's a happening place.
My best suggestion is to figure out how to navigate the city by the buses and subways. Rome's a place that you can walk yourself to death in. The buses are just so easy and efficient to deal with.

My preference is to spend that first day outdoors, not do too heavy a tourist site, and try to orient myself. The perfect thing would be to walk about, check out some places to eat, figure out the main routes.

We have been to Rome several times, a couple times for a week each (for business mostly) and stayed in the Trastevere. If I were arriving in Rome, my first day would be:

Follow Viale de Trastevere, the main street, across the river and work your way to Campo di Fiori. Hit one of the shops for some pizza by the slice to hold you over until dinner (try Forno Campo di Fiori) then stroll on up through Piazza Navona, by Trevi Fountain, The Parthenon, the Spanish Steps, basically Ricks walk through old Rome. Stop for a caffe if you get weary, find some gelato if it gets warm, a glass of wine if you want to sit for a while. Aside from food and drink, the cost is nothing. We also try to stop someplace and get some bottled water, some wine, and whatever else for the room, more if we have a fridge.

Head back to your hotel late afternoon, freshen up, do not nap, then head out and explore the Trastevere and scout for a place to eat.

Some of the places we would make sure to hit:
Popi Popi http://www.yelp.com/biz/popi-popi-roma Not fantastically the best, but very very good, nice selections...good Roman dishes, very reasonably priced, just a neighborhood place, can get lively on some nights.

Dar Poeta http://www.darpoeta.com/ they are getting more well known, but fantastic wood fired pizza, bit more variety of toppings than most, and you have to have the dessert calzone with Nutella and Ricotta. There are a number of other well known pizza places in the area as well, but this is the best, the others maybe more traditional.

For something a bit more upscale, and maybe not the first night, but we loved Ristorante La Scala http://www.ristorantelascala.it/

Aside from that, there are all kinds of little places scattered in the side streets. Do some looking on the web to see what is current and when strolling check out menus for what looks good. Piazza Di Santa Maria is also a good place to just sit and watch the scene as well as some decent restaurants.

The goal of the day is to get lots of fresh air, some exercise, figure out where you are, all to keep jet lag at bay. Hold off until 8:00 to eat, and when you finally hit your bed, you will sleep like a baby and be ready to go the next day.
Above all, have a great time