I suggest downloading the BKK map that has all transport lines that run on rails in the City from here.
Metros
Line 1.
Line one has no surface section. The trains emerge from the tunnel at the yard at the end of the line at Mexikói út. It is possible to photograph trains waiting in the yard through the fence.
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Another nice photo location is at the other terminus at Vörösmarty tér, where the trains change direction using non-revenue track.
Line 2
This line has 2 surface stations, Pillangó utca and Örs Vezér tere
West end of Örs vezér tere station is ideal for taking shots off arriving trains, and the yard.
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Both end of Pillango utca station are viable locations. The eastern end is ideal for taking shots off arriving trains, and the other end of the yard, while from the western end the tunnel portal is visible.
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Line 3
This line has only one surface station, the Southern terminus, Kőbánya-Kispest. It is possible to take photos of incoming trains from the end of the platform.
The only chance of taking pics of trains on the surface on the open line is from a bridge ~10mins walk from the station.
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Line 4
This line also doesn't have a surface section. The trains exit the tunnel just before the Southwestern terminus at Kelenföldi pályaudvar. Shots can be taken from the roof of a parking garage right next to the yard. It is not possible to take pictures through the fence of the yard.
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Over-under location
These locations are ideal for Tram and train over-under shots.
Railway line over tram line 17/61 on Villányi út
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Railway line over tram line 1 on Szeremi utca
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Railway line and tram line 1 passing over tram line 2 and 24 at Közvágóhíd
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Railway line passing over tram line 1
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Railway line passing over tram lines 17/41/47
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Railway line passing over H5 suburban railway line at Aquincum station
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Tram shots
Bridge or Parliament in the background while entering tunnel on line 19/41 at Clark Ádám tér
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Parliament in the background for line 19/41 at Batthyányi tér
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Line 2 near the Parliament
Multiple shots are possible as the tracks wrap around the building.
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Line 2, just south of Kossuth tér station, as the line joins the Danube bank the Castle Hill is in the background
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Line 2, exiting the tunnel under chain bridge
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Line 2, emerging from the tunnel south of Fővám tér station and passing some new/old buildings
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Railways
The biggest railway stations in Budapest are Keleti, Nyugati, Déli (these are terminuses) and Kelenföld (this is a through station)
Keleti pályaudvar
The trains leave the train hall to the east, hence the light is ideal in the moring to shoot departing trains. The platform between tracks 7 and 8 is the longest, hence walking to the eastern end of it allows for pics of trains leaving other platforms. In the afternoon hours it is also a good location to catch arriving trains.
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Just after leaving the station the tracks pass under Kerepesi út, there is enough opening to take pics of trains from here in the morning hours
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