Budapest is a walkable city. The vast majority of tourist sites, restaurants, bars, and thermal baths are easily within walking distance. With a little over 3km (1.9 miles) from downtown, one of the more far-flung destinations is the popular Széchenyi Thermal Baths, but the grand Andrássy Avenue leading to it more than compensates for the 40 minute stroll.
When crossing 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 375m/1,230 ft, and hiking trails are leading up to both the Castle and the Gellért Hills. Remember, walking is the best way to get a first-hand experience of a new city.
An alternative to walking is using BuBi, Budapest's bike-sharing system (this mostly applies to the Pest side; hilly Buda is a different story). The network, with 124 docking stations and 1,500 bicycles, provides an excellent coverage of the inner sections of Pest, primarily within the Grand Boulevard.
You will have to quickly register by providing your phone number and email address to purchase tickets at one of the main docking stations (dots filled with red) for a one day (HUF 500 or ~€1.7), three day (HUF 1,000 or ~€3.2), or weekly ticket (HUF 2,000 or ~€6.5). 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 of the city). The cost of a single fare is HUF 350 (little over a euro). One day, three day, and weekly tickets are also available. Tickets can be purchased at subway stations and major bus stops which all have vending machines that accept cards as well. 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. For the full experience, take it from Jászai Mari tér station to Fővám tér. To see the new Millennium City Center with the 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 a bunch 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 to operate 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 key points of attraction such as the Hungarian State Opera House, the House of Terror Museum, Heroes’ Square, and Széchenyi Thermal Bath. Hop on at any of the stations along Andrássy Avenue for a few stops to experience this charming piece of living history. You can learn more about the Millennium Underground here.
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 serene Buda hills. Take #105 from Deák Ferenc Square to Gyöngyösi utca for Angyalföld, and to Apor Vilmos tér for the Buda side.
Cab prices are regulated across all taxi operators in Budapest. The fare consists of a base fee of HUF 450 (~€1.5) plus a distance-based charge (HUF 280/km, or ~€0.9), 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 by the official prices displayed on one of the rear doors.
All cabs are required by law to accept credit cards. 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, such as City Taxi (+36 1 211 1111), Főtaxi (+36 1 222 2222), or Tele 5 (+36 1 555 5555) and tell them in advance if you will be paying with card. The driver will identify you by the name the car was called for before letting you in (there're no additional charges for calling a cab).
Thanks to the pressure the taxi driver lobby exerted on the government, Uber was banned in Hungary in 2016.