What's the best way to travel around Greece? Train or Bus?

It depends on the way you like to travel, the place you want to go and the budget you want to spend. When I was younger I used to travel around Greece by train, but the last few years I don't feel that safe doing so.

Also, the schedules are fewer than those offered when choosing a bus. Sometimes taking the fast train (intercity) might be at the same price it would cost you to get a seat on a bus or even catch a budget flight. There is no straight answer to this one. It has to do mostly with your destination and the time of your visit... Budget is also an issue. You can find a train ticket at the 1/3 of the price of a bus ticket if you are lucky to get an economy seat.

Although it is nice to travel by train I choose the bus most of the times since it can take me to most places. To get to the islands you might wanna do a bit of island hopping, you can check out the ferries section at Hostelbay for the Greek Island Pass if you like.

International trains to and from Greece have been suspended (see here and there). The easisest way to travel from Greece to Turkey will be by bus. There is also a possibility to travel by ferry.

I would not buy the railpass. You would need to take a lot of trains during these five days so that the pass would be profitable. Moreover, as the rail network is not very dense, you will need to supplement it by bus journeys anyway. Or you will miss a lot, such as e.g. the Peloponnese.

I suggest that you first make a list of where you want to go, and then check how it can be done. And then buy tickets (train, bus, ferry, ...) on purpose. With 150 USD you can already get far in Greece and see quite a bit.

A more general remark. Rail passes may look interesting at first sight. However, this may not necessarily be true. It is always interesting to compare passes to point-to-point tickets. You can refer to a discussion on this site or to seat61.com.

Even though I am Greek, I haven't lived there for a long time. Last time I traveled to Greece I also went via Istanbul and took an overnight train from there to Sofia and then a plane down to Athens. Before that, I have take trains between Athens and Thessaloniki (the second largest city) and Athens and Patra (another large city). If you are backpacking and want to see the most beautiful parts of Greece (in my opinion Crete and the Islands and the mountains of central Greece and Peloponnese) then you are likely to need to take other means of transportation such as ferries, large boats and buses.

I really like trains and dislike buses but the few times I have taken an overnighter in Greece and in Turkey the conditions were like getting yourself in a train for the front in WWII (In fact once I was traveling with soldiers as they get free tickets on the overnighter I think). Both in Greece and the one to Sofia from Istanbul were very slow, not very nice and not very comfortable. I know that the day trains are much better and faster than the all night trains, but I haven't used them much (as back when I was travelling around Greece I was going for the cheapest).

Based on those experiences I can recommend you that, depending on your budget and your travel plan, and whether you dislike buses as much as I do, try the following in order of preference:

1. Get a day train from Istanbul to Thessaloniki if there is one and it looks decent, and to the same between Thessaloniki and Athens and Athens and Patra.
2. If you can afford it, fly to the furthest away islands and to Crete.
3. Definitely take the "odondotos" (gr. Οδοντωτός : rack-and-pinion railway). The area is beautiful and the trip rewarding. Info here: http://www.odontotos.com/index-en.htm
4. Don't get a railpass for Greece only for the price you mentioned it, but look into Eurail passes (3 days in Greece for 102 euros http://www.eurail.com/eurail-passes/one-country-pass/greece). The reason for that is that you are unlikely to need to travel 5 days on rail.
5. Take at least one boat trip to an island.
6. Have a great time in one of the most beautiful corners of the earth :-) Cheers