Disposable take away cups are difficult to recycle due to the mixture of materials they are formed from. The main issue is they usually have a thin plastic lining on the inside which is difficult to separate from the rest of the cup for recycling.
Carrying your own cup will reduce the need for disposable cups and will in some coffee shops save you a little bit of money as they offer a discount for using your own cup. If you regularly get a drink while I this amount will soon add up.
There are a whole variety of cups available in different sizes and styles. Your choice of cup will depend on you budget and requirements. It is a good idea to look for one that offers good insulation as this will help to keep your drink hotter for longer. The type of lid can be important too as this will be important in reducing the risk of leaks and spills when moving around.
I have a reusable cup that is mainly made from plastic however I have had it for about 10 years and it is still going strong. If it becomes unusable, I will look for an alternative made from a more sustainable alternative possibly a glass or bamboo one.