Although I would usually recommend travelling by coach to Eger, there are times when the train is the better choice.
The biggest problem I have with the trains is that it is a bit of a lottery as to what type of train you're going to get, sometimes it is a spanking new red one and at other times, something that looks like it hasn't been cleaned since the 1950s greets you.
Some of MÁVs trains have seen better days
Trains to Eger are relatively frequent and leave from Keleti train station (Keleti Pályudvar) which is can be reached by the M2 and M4 metro lines.
You can access the the English timetable here. There are two option open to you: firstly, you can take a direct train from Budapest to Eger (leaves at 07.00, 09.00, 11.00, 13.00, 15.00, 17.00) or secondly, an intercity to Füzesabony and then change to a local train; this option take 5 minutes longer but at least you are guaranteed a comfortable ride most of the way. You will, however, have to wait at Füzesabony for up to 20 minutes
As for ticket prices: the first option is 2725 huf one way and double that for a return, the second is slightly more expensive at 3025 huf, double for a return. (prices correct as of January 2015 - please check the MÁV website for current fare)
The main Eger train station
Tickets: When travelling to Eger by train, you have to buy your tickets at the station or on-line. You will need to specify the time of the train you are travelling on as tickets are valid for specific journeys (unless a return).
It is not possible to buy tickets on the train and if the ticket inspector finds you travelling without one you could end up paying a hefty fine. No amount of remonstrating will help, as he/she is unlikely to speak any English.
Seating: Intercity trains require seat reservations (included in the ticket). On other trains you only need to reserve a seat if you are travelling in 1st class. On Thursdays and Fridays the transport network is busy and if you want to be guaranteed a seat then you should aim to arrive half an hour before departure time. If you need to buy a ticket before embarking ensure you arrive with lots of time to spare as the queues at the ticket offices can get pretty long, particularly on the aforementioned days.
Refreshments: If you're travelling with children then stock up on snacks/drinks as you won't get them on the train.
Bicycles/Dogs: Special tickets need to be purchased for bicycles and dogs. Most trains have an area set aside for bikes. Dogs must have a muzzle and be kept under control for the duration of the journey.
Disabilities: For those with disabilities, I would advise against travelling by train to Eger. Due to the age of the trains (even the Intercities), there is little if any provision for those not able-bodied. Going by coach would be a better option, or taxi. We can arrange taxis from Budapest to Eger for half the price you will be quoted by firms in Budapest. Please contact us if you need to avail of this service.