I will be going to Budapest in October for the first time. Just wondering how many days we should stay to see the city?
I'd stay forever if I could as well! It's THAT awesome of a city! You can get to most of the sights and experience enough cool things to do in the city with 4 days. But a week will give you more time to see other things. If you can hire a car, I'd plan a day trip somewhere like one of the smaller "major" cities like Miskolc or Debrecen. Miskolc is close to the Tokaji wine region so if you're a wine fan that might be a great day trip.
An interesting town that's definitely off the beaten path is Sumeg. It's a small town southwest of Budapest with a castle situated on top of a huge hill.
I would argue that there are a few cities more worth visiting than Miskolc (which used to be an industrial city) and Debrecen (a bit far away in the Great Plains). If you want to venture out from Budapest and not go far away, I can recommend the Dunakanyar ING, which features some nice places such as Szentendre, a small village famous for its art scene and more importantly a great open-air folk museum Szentendre Skanzen Village Museum that is definitely worth a visit (plan for a full day trip). There is also Visegrad , also a small place with a castle on the hilltop. On the other side you can visit Godollo, with the residence of the famous Sissy.
It's really funny that of all places you have mentioned Sumeg, it is actually my hometown :)
It is really a nice little place to visit, but I would make it part of a bigger trip along the Balaton Uplands National Park.
as for Budapest, seriously plan at least a week. Even if you never leave the city it will feel like not enough.
If you are travelling between May and September and have the chance, try to go on a roadtrip to the Lake Balaton area for a week, maybe from Siofok (summer party town on the south side) going west until Keszthely. You can easily drive in a couple of hours, but if you are more adventurous you can rent a bike and make use of the cycling path that runs around the lake. Along the Balaton these places are nice to visit: Balatonfured (one of the major cities, nice city beach), Veszprem (county capital), Tihany (lovely village on the peninsula), Badacsony utca (major vine town of the area), Szigliget (picturesque medieval castle & lesser known beach), Tapolca (lake in a cave where you can take a boat to go around, lovely small town centre), Heviz (Unesco world heritage open-air thermal lake), Sumeg (medieval castle with daily live performances during the summer season), Keszthely (beach, castle, tiny quirky museums...).
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Budapest is an awesome city, I'd stay forever if I could :) I would give yourself a week at least. October is a good time to go since it won't be peak season with massive crowds.
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4-5 days are enough if you only plan to visit the most popular attractions of the city
Have a nice trip.