There are numerous ways to travel in Turkey. It all depends on what you need. Whether you plan to take it slow or try to get in as much sightseeing as possible, both basic and luxury options are there. There's something to suit everyone.
The best way to explore towns during your visit is by foot - take your time, wear comfy shoes and go for a stroll. In Istanbul , for example, the scene and atmosphere changes with every block. To cross longer distances, taxis (or taksi in Turkish) are an option. To avoid scams, only use licensed taxis and make sure the meter is on. And the driver must set it to zero before you get in.
There's also the cheaper Dolmus minibuses, which are slower but you get to travel as locals do. They are shared taxis, so you'll find yourself squished in with other customers. Their routes are fixed, so make sure you tell the driver where you want to go - otherwise you could end up heading in the wrong direction.
In Istanbul you've also got a subway, trams and cable cars. As the town is divided by the Bosphorus River, you'll probably use a ferry or taxi boat at some point.
Turkey is vast and distances between towns are often huge. Fortunately the road system is well maintained.
Traveling by bus is the usual way of getting around. It's relatively cheap and, if you want a bit more comfort, you can always go First Class. It's best to reserve your seat in advance, as it's a popular way to travel and tourist destinations are quickly booked up.
As for trains, you need to be a fan, on really tight budget or have a lot of time on your hands. They are really cheap. And really slow. Bus is generally best.
If you're in a hurry and want to see as much as possible, you can always fly. Low-cost operators offer competitve prices and you'll save a lot of time, compared to traveling by bus.