train from Izmir to Istanbul

Not exactly an answer to your question but there are quick inexpensive flights. Try Atlas Jet, Onur Air, or Pegasus.

I'm not sure about the train. Trains are not used as much in Turkey as other European countries. There are flights as already noted. They will take about an hour plus before and after time at the airport. There are buses if you want a cheaper or non-flying option. I'm not sure about the length of the bus trip, but would imagine its 8 to 10 hours. Good luck.

if you can't find a cheap flight you can take an overnight bus. leaves around 10 arrive in IST around 6. no need to book ticket in advance, just buy it at the bust station. you can use the pamukkale bus company, big red bus you can't miss.

There currently isn't a direct train. You will have to change at Bandirma, and you might have a number of hours to wait for the Bandirma to Istanbul connection. Discount plane tickets will be only a few lira more than the train or bus, which are both comparable in price. The train route, as I recall from doing it many years ago before the rise of cheap airlines, is striking in that it goes along the sea for a while. Can't speak for the bus, but the bus route will involve a ferry passage on the way. Planes are quick (about 45 minutes wheels up to wheels down) but there's the time of getting to the airport, getting through security, and getting back out again. If you go by plane, assume that you will spend a good part of the day in transit. If you go by train or bus, assume it will take a whole day (or night) in transit.