Please excuse my ignorance!
To start with, please recommend a decent hotel in Istanbul, likewise in Izmir.

We are planning just over 2 weeks in Turkey over Christmas and would appreciate any ideas on what to do there, we mainly want to take in archaeological sites.

Thanks in advance

When we went to Turkey the first time, there were cheap flights to Izmir. For archeological sites, you want to stay in Selchuk. I wrote a trip report which mentions hotels and sights. Click on my name to find it.

A car would be useful. You have plenty of time for visits. I think we traveled for nine days to see Ephesus, Pamukkale and Cappadoccia, admittedly with a couple of long days driving here and there.

Why it's a good idea to visit Ephesus before 10a.m.:

https://flic.kr/p/7pU6eF
Bring a swim suit and you can paddle in a hot pool among the ruins:

https://flic.kr/p/7pUdRP
Here is the album of our ruins trip:

https://flic.kr/s/aHsjpcrHU1

We have stayed at Hotel Uyan in Sultanahmet, the old city, were there again last month. The Uyan is a small boutique hotel, nothing fancy but neat and clean, with helpful staff and delicious breakfast. Gorgeous views of the Blue mosque and Aya Sofya from the rooftop, we took our breakfast up there a couple of times. We walked everywhere, all the historical sites are in the area, the tram line is also close by.

There’s a discount if you book directly on their website, I believe we paid a little less than €70 for a double room.

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We have good coverage and mapping of the archaeological sites at The Megalithic Portal - https://www.megalithic.co.uk/article.php?sid=13596

I suggest you have a browse and decide on which sites you want to visit. If you have a car, it makes it a lot easier. But negotiating a price with a taxi might be a good option. For instance a trip from Selcuk to Claros, Metropolis, Magnesia and back is 132km which would cost about 50USD (at 4TL per km taxi rate - https://taksiyle.com/en/izmir).
Google maps is also good for planning, most of the ancient sites are marked - https://goo.gl/maps/M5e9T27asWCMJPh78

It’s been 20 years since we were there, but we stayed in the Mavi Ev (Blue House) hotel in Sultanahmet in Istanbul and enjoyed it very much.

From Izmir you can also visit ancient Pergamon.

Main archeological sites around Izmri:
Pergamon, Pamukkale/Aphrodisias, Ephesos, Priene, Didimea.
Main archeological places aorund Antalya (usually better climate in December):
Perge, Aspendos, Side, Phaselis, Termessos

Here's our TR from our 25 days in Turkey in 2013. We thought the Dersaadent Hotel in the Sultanahmet area (http://www.dersaadethotel.com) very nice. I have heard from others more recently who stayed and agreed.
The Road to Ruins - 25 Days in Istanbul & Road Tripping thru Western Turkey