Easy steps to find accommodation in Rome:

The best choice for a first-time visitor is to stay downtown (like near the Pantheon): most attractions are walking distance from there, it will save much time from transportation and leave more for enjoying the city. Hotels in the downtown are costly but you can take advantage of many online offers

A nice self catering apartment is a decent alternative, especially for couples and if you don't mind cooking yourself from time to time: it will save even more of your budget. You may want check apartments availability here:

You can also see all available rooms online: