You want to be central, because then you can walk everywhere. I would suggest the pantheon area or around piazza navona. It’s tempting to save money by staying further out but trust me, it’s not worth it. From those central areas you can walk to the colosseum in one direction, and the Vatican in the other.
However, please please book entrance tickets in advance. You will regret it if you don’t because you’re then likely going to have to queue for ages. There are some really good tours you can book, which if you are interested in history and/or architecture (and I’m be questioning why you are choosing Rome if you aren't) are a must. There’s a brilliant tour the Vatican runs that you have to book way in advance. It’s run by an archaeologist and you can go deep under the Vatican. Castle S’ant Angelo is another of our favourites. As well as the Capuchin Crypts. And the Roman Forum is sublime.
Get a good guide book, but don’t overlook just popping into churches at random because you will see some incredible art work.
We often go in January (we included it as part of our honeymoon one time, along with Florence and Venice stays - it’s very easy to use the trains in Italy as they are so much better than ours, so we did all three places on that particular visit). January in Rome tends to be cooler but sunny (we’ve yet to have rain on a January trip). There are usually a lot fewer people there.
I would also be careful about choosing where you eat. There is a lot of touristy crap but there are also some amazing places, but you need to be careful.
I love Rome. It’s the most amazing place. I could talk about it all day. I keep going back and finding new things to do. But I find you need to research and plan to get the most out of it. I lost count of the amount of times we heard people complaining about wait times or lack of things to do because they didn’t bother looking at anything in advance. People also often neglect Rome by only going there for three or four days. Don’t make that mistake. There is so much to do.
The very first time we went, we booked a night tour of around 90 mins around Rome just to familiarise us with some of the area. It was brilliant, and quite dark (ghosts and mysteries tour).
Happy to give more info on anything if you want.