Airport Shuttle Bus (100E) / Taxi
Budapest has an airport shuttle bus (100E) that connects the city center (Deák Ferenc Square) with the Ferenc Liszt International Airport. It's a quick (~30 min) and cheap way to get in and out of the airport by public transport. A one-way ticket costs HUF900 (€3) and can be purchased at the stations and the airport. You can pay with a credit card if you use the vending machines, otherwise pay cash (in Hungarian forint) to the driver.
The airport shuttle buses run every 20 minutes. The only downside is that they can get uncomfortably crowded. When heading from the city center to the airport, I recommend that you get on at the Deák Ferenc Square station as you're more likely to get a seat, and the station there has a ticket vending machine that accepts credit cards too (the Kálvin Square stop is cash-only, local currency).
If you decide to take a taxi, note that there's no flat rate service to and from the airport. The licensed cabs (see more below) will use a meter, and depending on your final destination the fare will usually be anyhwere between the equivalent of €25 (HUF7,500) and €35 (HUF10,500).
Exploring Budapest - Walking
Walking is the best way to discover a new city and Budapest is easily walkable - most tourist sites, restaurants, bars, and thermal baths are easily within walking distance. One of the more far-flung destinations is the popular Széchenyi Thermal Baths. It's a little over three kilometers (1.9 miles) from downtown, but the 40 minute stroll through the grand Andrássy Avenue is worth every minute of it. This district-level overview will help you familiarize yourself with the main pockets of Budapest.
When you cross over to Buda, don’t give in to the mental barrier that the other side of the Danube is too far. The bridges are shorter than you think, the Chain Bridge is only 375 meters, or 1,230 feet, and hiking trails lead up to both the Castle Hill and Gellért Hill.
An alternative to walking is BuBi, Budapest's bicycle sharing system. With 126 docking stations and 1,500 bicycles, the network provides an excellent coverage of the inner sections of Pest, primarily within the Grand Boulevard. Although there are docking stations in Buda too, its rolling hills are less biker-friendly than the flat Pest side.
To rent a BuBi bike, you will have to quickly register by providing your phone number and email address at one of the main docking stations (dots filled with red) or online. One-day (HUF 500 or ~€1.5), three-day (HUF 1,000 or ~€3), and weekly tickets (HUF 2,000 or ~€7) are available. The system will charge you a fully-refundable security deposit of HUF 25,000 (~€80) that's automatically released after your ticket expires. The bikes are free to use for 30 minutes at a time, after which additional fares apply. With plenty of docking stations scattered around the city, you shouldn’t need to exceed the half-hour limit unless by conscious choice.
Budapest has an especially good public transportation coverage. Buses, trolleys, or trams roam most parts of the city (subways too, but then you don’t see as much). The cost of a single fare is HUF 350 (little over a euro). One-day, three-day, and weekly tickets are also available. You can purchase tickets at subway stations and major bus stops which all have ticket vending machines that accept credit cards too. You can also buy tickets directly from the bus drivers, but those will be slightly more expensive and you will hold up traffic. For directions and schedules, you can rely on a most loyal friend: Google Maps.
Several public transportation lines in Budapest provide a unique sightseeing tour for merely the cost of a fare. On tram #2, you will have plenty of opportunities to marvel at the Hungarian Parliament Building and the Buda Castle, a UNESCO World Heritage site, from along the bank of the Danube River. Take the tram from Jászai Mari tér station to Fővám tér. To also see Budapest's National Theatre and Müpa, the home of the Ludwig Museum of Contemporary Art, take tram #2 all the way to Közvágóhíd.
Trams #4 and #6 travel mainly along the Grand Boulevard, the dividing line between inner and outer Pest. With plenty of sites along the way, take your journey from Széll Kálmán tér in Buda to Boráros tér in Pest.
The Millennium Underground of Budapest began operations in 1896, making it the first subway line on the European continent (today it's also a UNESCO World Heritage Site). It runs mainly along Andrássy Avenue and passes several points of attraction like the Hungarian State Opera House, the House of Terror Museum, Heroes’ Square, and Széchenyi Thermal Baths. Hop on at any of the stations for a few stops to experience this charming piece of living history (here you can learn about the Millennium Underground).
Bus #16, also known as the Castle Bus, will take you from the city center at Deák Ferenc Square up to the historic Castle Hill in Buda. It crosses the Danube through the Chain Bridge, the first permanent bridge connecting the two sides of the Danube in the city. Get off the bus at Dísz tér, and you're ready to roam the streets of the Old Town.
A trip on bus #105 offers a different, but similarly unique experience as it passes through the pristine inner city with one endpoint in the working class neighborhood of Angyalföld, the other in the verdant and prosperous Buda hills. Take #105 from Deák Ferenc Square to Gyöngyösi utca for Angyalföld, and to Apor Vilmos tér for the Buda side. Sit back, relax, and take in these wildly different faces of Budapest.
Cab prices are regulated across all taxi operators in Budapest. The fare consists of a base fee of HUF 700 (~€2) plus a distance-based charge (HUF 300/km, or ~€1), irrespective of the time of day. Be wary of unlicensed cabs to avoid being overcharged, although now they're less common in Budapest. Licensed cabs are yellow, and can be recognized by the corporate logos on both sides of the front doors, and the official prices displayed on one of the rear doors.
In theory, you should feel free to hail any yellow cab off the streets without being overcharged, but it’s safer to call one from a reputable company like City Taxi (+36 1 211 1111), Főtaxi (+36 1 222 2222), or Tele 5 (+36 1 555 5555). All cabs are required by law to accept credit cards.
I generally avoid freelance cabs. They're also yellow, but instead of the corporate logo, "freelance" and "független szolgáltató" are written in black on their front doors.
Thanks to the effective lobbying efforts of local taxi drivers, Uber has been banned in Hungary since 2016.