Most of the sites in Budapest are within walking distance. 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 walking is the most efficient way to explore a city.
If not walking, using BUBI, the bike-sharing system of Budapest is one of the quickest ways of getting around in Pest (hilly Buda is a different story). The network consists of a total of 112 docking stations and 1,300 bicycles, and provides an amazing coverage of the inner sections of Pest.
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 1 day (HUF500 or ~€1.7), 3 day (HUF1,000 or ~€3.2), or weekly ticket (HUF2,000 or ~€6.5). The system will charge you a fully-refundable security deposit of HUF25,000 (~€80) that's automatically released after your ticket expires. Up to 4 bikes can be rented with one user ID. 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 30 minute limit unless by conscious choice.
If not walking or biking, public transportation is the way to go. Budapest has a particularly good coverage by public transportation, including buses, trolleys, and trams (subways too, but then you don’t see as much of the city). 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 trem #2, you'll have plenty of opportunities to marvel at the Parliament and the Castle District, a UNESCO World Heritage site, from along the bank of the Danube. For the full experience, get on at the Jászai Mari tér station and take the tram to Fővám tér. If you're also interested in seeing 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. To see the best bits, take your journey from Széll Kálmán tér in Buda to Boráros tér in Pest.
The Millennium Underground of Budapest began to operate in 1896, which made 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 Opera House, the House of Terror Museum (Vörösmarty utca stop), 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, or the Castle Bus, will take you from Deák Ferenc tér in Pest up to the historic Castle District in Buda 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 historical old town.
A trip on bus #105 offers a different, but similarly unique experience as it crosses through the pristine inner city to terminate on one end in the mostly working class neighborhood in Angyalföld, and the peaceful Buda hills on the other. Take #105 from Deák Ferenc tér 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 HUF450 (~€1.5) plus a distance-based charge (HUF280/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, as well as by the official prices displayed on the outside of 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) or Főtaxi (+36 1 222 2222), and tell them in advance if you'll be paying with card. The driver will identify you by the name the car was called for before letting you in (there are 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.