what is the best time of year and best route to travel the mediterranean countries at low budget as a single?

I want to see Greece, Israel, Egypt on low budget. I am a single woman traveller. Please advise with respect to places to stay, mode of travel to be taken within the places, time of year

Jan

January?? Are you kidding? It's winter there and a lot of the mediterranean places are closed over the winter. Make sure you travel out of school holidays.
Personally, I think cruising (as in, on a boat) is the best way to see this area, but it is a personal choice. Check out these websites:

http://www.greecemedtravel.com/greecemedtravel/default.asp
http://www.medtours.com/
http://www.costacruise.com/B2C/USA/Default.htm
http://www.mtfcairo.com/App/homepage.cfm?moduleid=3140&appname=100481
http://deals.virtualtourist.com/viewDeal-6144-0-1---viewDeal.html

I think the best time is late summer ideally September. Its still really warm but a little less busy. Go Island hopping in Greece. The most beautiful Island is ,I think, Symi in the Dodecanese.You will be ok alone as long as you keep your wits about you. Have fun!

Egypt is fantastic went august it was hot but but not humid they need tourists and you will get spoilt by Egyptian males and the rest.. luxer go for cruise down Nile and visit valley of kings and queens fantastic book tours when you get there cheaper .. have fun

I lived in Egypt for a while and I have been to greece as well. both really nice, take the end of the summer as said above from middle of september to october. less tourists and therefore cheaper. but dont go to late, they got a pretty bad winter too.

Nile cruising is a nice idea and so is boating the islands outside greece. ive heard its cheap to go with boat from egypt to greece so I would recommend boat as your travel vessel. however, start the trip in the country that is cheapest to go to. and go to the other by boat. good luck miss!