Everyone's trip will be different and for varying lengths of time. I am going for a few months so I'll be including travel for both longer and short visits.
Each of these methods has its benefits. Taking a train is great for a more scenic tour of the cities or countryside. The downside is that it can take some time. Not as much time as say a bus or cab, though. Planes are often the fastest option but also can be the most expensive choice. Another factor in planes is that airports are usually a good distance from where you will be staying.
Traveling by plane is relatively easy and saves time. There are a few factors to consider here; the first being money as it can get expensive, the second is your transportation to and from an airport which can also add up. Depending on where you are flying you can get a flight with Ryanair, EasyJet or WOW for about the same as a bus. When doing research these airlines don't often show up on sites like Kayak. I found a site Kiwi that has a great search function. You can search a destination or even a general area and then set a date or range and It will list the best flights. Also best to book when you are ready to fly not from home.
This is not a bad budget option. You will likely meet some fun people or some not so friendly ones. The only real problem to consider here is time. Bus is by far the slowest and most scenic option. I should note that if you are planning to travel between countries this is not the option. Cross countries is both very time consuming and will cost more than a plane.
This is the method most tourists use. This is also the option that is most advertised and seems to be the mode of transit around Europe. This choice more than other can vary greatly depending on length of stay and what time of year. You can get a EuroRail pass for varying lengths of time and the price is determined by days of travel in any given time frame and the number of countries you are traveling. I should note here that your EuroRail pass only includes travel for a number of days, unlimited during those days. Also if you plan to travel overnight ( to save money on a hotel) you will have to book in advance and pay a booking fee which can be pricey up to 100 Euros each time.
I think overall for a trip of a few months your best option is to do a blend of each method. It's a great idea since you will get scenic, fast and tourist options. If you are taking short stints or to just one country a train maybe the most fun and economical. Within a single city just buy yourself a transit pass for your time and not have to worry. If you are trying to plan a trip and don't have destinations or length in mind GoEuro is a fantastic tool to use. It allows you to enter destination and travel date and will then search bus, train, plane and even car rental and give you the prices for all of them and book.