Well to get to ancient Greece and Rome, first you have to find a time machine...
Seriously, there is a wealth of information on the internet on traveling to these two place, any info you get here can't equal what you find on your own searches. Also go to the book store and get a travel book or two - Frommer's or, I like the Lonely Planet series for in depth indepedent travel with lots of info on hostels, etc. I also like Rick Steve's series of travel books. I would choose one or the other (Greece or Rome) on a, I assume, 2 week or less vacatiion, not both. But to give a brief summary:
Best time of year to go: Summer in Athens is HOT, uncomfortably hot. And also crowded with tourists. Winter is good - no tourists, not too cold, and air flight prices from the U.S. is really cheap. But the days are shorter and some of the sightseeing spots are closed or have reduced hours. Fall and Spring are the best. You might have to tend with rain in the Spring.
How far in advance to get your tickets - doesn't really matter as long as you don't wait longer than 3 weeks before the trip. Look out for sales and grab them prices, if you don't like the current prices, wait, they will drop.
Affordable hostels - see advice on searching elsewhere and check out Lonely Planet, etc. Lonely Planet and Rick Steve's also has a web site (do a search). You won't get more than anectodal info here. Both Rome and Greece have a subway system, not the best, but they have it. Look for hostels near these.
Rental car services - If you just visit Rome and Athens you do not need nor do you want a rental car. Both countries have good train service as well. I traveled up and down the boot in Italy by train, in Greece they have an excellent ferry service if you want to visit the islands.
Rome and Italy was one of my favorite vacations and Greece was great - spent about half the time in Athens and about half the time in the Agean Islands.
and tour guides - bleh! I avoid them and like the independent travel. You will have more fun with doing your research and ditching the tour guide cattle movers.