Get a Lonely Planet. Or a few digital chapters from their online store. Probably the best place to start. Wikitravel is a good resource, but lacking on more obscure destinations.
Unless you have boatloads of time and assuming you're traveling overland, skip Egypt. There's no easy way to get there from the northern mediterranean, unless you fly.
Buses in Turkey are very affordable and good. So are trains, but their network is limited. You can assume that it will be easy to find busses from larger cities to larger cities in Turkey once you get there.
Trains in Greece ain't half bad and affordable. The ferry from Greece to Italy can be pricey, so compare your options. For ferries, try ferrylines.com.
Slower trains in Italy are well priced.
Budget airlines serve several cities in Turkey from central and south eastern Europe. Check hipmunk.com and AirNinja.com to look for cheap flights and budget airlines respectively.
Use hotelscombined.com, bookhostels.com and airbnb.com to find affordable accommodation.
For international train travel, start at seat61.com.