If you really prefer not to fly the only practical way is by bus. These are comfortable and swift but cost only a little less than a flight.
As an example, METRO have buses every couple of hours. The journey takes 9 hours and costs 90 TL.
Depending on the date, you may find flights for about 110 TL.
The train is the cheapest, but not a great option unless you're an enthusiast. You would start with a 2 hour ferry ride out of Istanbul, and works are still going on along the line.
Annoyingly, the METRO search box doesn't recognise the word KUSADASI. You need to scroll down and find KUŞADASI (!)
Incredibly, the page then shows results for "KUSADASI". This is unusually poor computer programming from one of these excellent companies.
Apart from Metro I would recommend Kamil Koç.
Both companies have 3 buses every day serving the route Kuşadası to Bursa.