Budapest is safe. You'll be fine.
About the nightlife... Two words: Hajogyari Island. Yes, Budapest has a thriving bar and pub scene. Yes, it has an interesting art and music scene (more on that below). But if you want hot girls then you need to spend all of your nights out at Hajogyari Island. The island is about 20-30 minutes outside of Budapest by taxi. You can get there by public transit, but you have to make a few connections and it takes a good 45 minutes each way. The island is an island in the Danube and holds about half a dozen clubs.
I can honestly say, in everywhere I've been in the world, I have never seen a higher concentration of hot girls in a single place than Hajogyari Island. There are places where the girls are hotter (Ukraine and Russia), but never have I seen so many hot girls in one single venue at the same time. Particularly the night clubs Bed Beach and Dokko Club. You literally have to look for an ugly or fat girl. It's impossible to find one some times. A number of guys I've spoken to who have gone to Budapest and commented on how there were no hot girls out at night. I thought the same thing. It's because they all go here. Seriously. It's the only city I've come across where literally all of the hot girls congregate in a single place.
With that said, the clubs on Hajogyari can be brutal. They're some of the most difficult places I've tried to pick up. Loud club environments with large VIP sections. Large groups of people and lots of guys. Language barrier.
So why would you subject yourself to this abuse? Well, if you're like me, you get some sick thrill out of it. But the sheer quantity of hotties keeps you going. Every time you get down on yourself, you spot even more walking past you. Honestly, it's a crying shame that Hungarian is such a useless language, otherwise moving and setting up shop in Budapest and forming a social circle would be lethal here.
If you aren't up for running the pick up gauntlet, you can stay on the mainland and hit the bars. Most of them are in the Jewish quarter and near the Octagon.
Budapest is quite proud of their eccentric and odd bars. And I have to admit, the way they're decorated can't really be described. Very artistic, very interesting. Cool vibes. The bar I enjoyed the most was a bar called "Szimpla," which seems to be somewhat famous. It's fairly large and drew a good number of tourists and ex-pats as well. I met a Brazilian girl there and spent a couple nights with her.
As with Prague, the strip clubs are scams. Also, there's a well-known scam of girls stopping foreign guys on the street and inviting them up to get a drink with them. Once they get you up there, they order drinks without letting you know how much they are. The drinks end up being like $100 a piece and they won't let you leave until you pay. I didn't run into this, but I read about it (I think someone here even mentioned it). Just remember: if it sounds too good to be true, then it probably is.
As with anywhere in Eastern Europe, language barrier is a legitimate issue. I'd estimate that 50% of people speak some level of English, but most of them poorly. Many of them speak some German instead of English.
The women are friendly but it seems to be a bit of a social-circle oriented culture. The rejections were never harsh. But they were clear. It was hard to break into groups and the guys weren't exactly the most keen to meet foreigners.
Things to do: Check out the Basilica, the Opera House, the Parliament. There are some gorgeous parks near the river. Budapest is a beautiful city. Taking a walk over the bridge and to Magrit Island is a must if the weather is nice. Also it's a great city for a bike tour.
Budapest is also known for it's summer festivals. They have a ridiculous amount of them (like 50 or something) and they're going on all the time. Budapest is a lively music and art town, so there are tons of opportunities to check out interest jazz, folk, or blues music. There are salsa and dance scenes if you're into that. And there's an impressive array of restaurants to partake in. The food there was pretty good. Although I was usually too hungover to partake.
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