If you’re traveling by train in Eastern Europe and only stopping for a day in Budapest (which would be a pity!) then you might want to choose the Budapest-Keleti Train Station (Keleti Pályaudvar) as your luggage storage. The station is on the Pest side of Budapest and right upstairs from the M2 Metro line (Red Line) which can take you into the city center or even all the way across to Buda and the Southern Railway Station (Déli pályaudvar).
As you walk in the front doors of the station (on the south side of the train terminal) approach the tracks and turn left. Aroom of lockers is located in between the Lotto and Mini ABC shop/Customer Service office. Also look for another location under the main stairs, but there is no change machine there.
Storage lockers in Budapest-Keleti Station come in a variety of sizes.
All are coin operated, with sizes to accommodate small soft bags, through medium- and large-sized suitcases and even a proper tall locker like you had in high school maybe. Prices are 400, 600, and 800 forint. Machines accept 50-, 100-, and 200-forint coins. [Small lockers are approximately 45x80x35 cm; the larger approximately 65x80x55 cm costs 600 HUF/24 hour.]
One nice bonus is that some of the lockers even take Euro coins (price is €3). There is a change machine at the entry, and an ATM nearby as well if you have nothing to change.
Pack your bags into an open locker, close the door, deposit the coins, turn the key to lock it, and keep that key carefully on your person. You now have 24 hours to explore Budapest. If you go over 24 hours (up to 72 hours, though I don’t think they are policing that like they do in say Munich train station luggage lockers) you will need to add money, according to the digitally displayed balance, in order to open the locker with that key.
From Budapest-Keleti train station you can walk outside and head down into the Metro system at Red Line Metro Station Keleti Pályaudvar.
Reminder about the Metro: Whatever you do, be sure that you purchase the correct ticket for the Metro and VALIDATE it before you descend to the platform. Warren Beatty, in Budapest once for filming a movie, famously failed to do so and was smacked with a fine. So fat chance you’ll be talking your way out of one.
Be aware that if you are going 3 stops or less on the same line, you can get a slightly reduced ticket at the ticket window.
Tickets also come in 10 packs making them equally lower in price as the 3-stop Metro ticket. There are also some day passes if you plan to use the Metro system and buses numerous times in a day. Buy them at the station ticket offices or at a newsstand.