It is very easy to get around the enchanting city of  Istanbul. 

If you stay in Sultanahmet neighbourhood it is full of historical things to see in terms of museums and the famous Blue Mosque, Hippodrome, Hagia Sophia and Topkapi Palace (some require entrance fees between 10 and 20 TL). These are all in walking distance around the Blue mosque area, more importantly there is a state of the art and very frequent local tram line. It's 2TL per trip of any distance, you buy a token at one of the machines to enter the station. You should pick a hotel near the Blue mosque, you can see the location of the hotel on the maps when you're booking, there are many and they all seem to come with breakfast included, try to book a hotel with rooftop breakfast to get the most of the view which includes the Bosphorus/Marmara sea.

In terms of airport transfer, there are many, usually around 20 Euros each way for 2 people and you pay when you see them. Alternatively, for the adventurous,  you could use the Tram if you are travelling very light but it's recommended you try and find a map on-line beforehand to plan ahead as you may need to make station changes. At the airport you can pick up a free city map from the tourist information desk before you get in your cab.

You get more Lira for your GBP/Euro in Istanbul, and there are quite a few money exchange places so shop around.

Take the tram to Eminonu and there are many boats to take you for a tour on the Bosphorus, you don't need to pre-book. This is also the stop for the Egyptian Bazaar and a huge local labyrinth like market area behind this where you can buy everything from house wares, clothes to local produce, including a freshly made ground coffee outlet that you can bring home, Turkish delight and lots of nuts.

You can really enjoy Istanbul by foot, there are many cats and dogs around but they are very tame and friendly and seem to be well fed! There are countless souvenir shops and people offering to polish your shoes. Many restaurants and off course Kebab shops. Make sure you try the fresh juices made in front of you namely orange and pomegranate juices.

For more branded High Street type shops, you can get the tram to Taksim square and walk down the very long and lively Istiklal Avenue. Check the weather on-line before travelling as Istanbul can get very cold.