The two most famous among the tourists are the Szechenyi baths and the Gellert baths though you might prefer to go to the ones frequented by the locals (and which are a bit cheaper). All the details can be obtained from this website. Many of them open as early as 06h00 and remain open till about 19h00 or so every day of the week.
Despite its steep entrance fee (4,100 forints or 18 USD for the day) the huge Szechenyi thermal baths complex could be well worth the money as it is a once-in-a-lifetime experience, such as leaving a sweltering and steam-filled cabin where the temperature is over 70°C to plunge completely into an immersion pool with the water at hardly 18°C. But most people would rather just relax comfortably for hours in the dozen or so "medicinal" pools with temperatures ranging from 28°C to 40°C.
It is very easy to go to the Szechenyi thermal baths complex by public transport as it is just a stone's throw away from the metro station carrying the same name (Szechenyi furdo in Hungarian). When you come out of the station you'll see a huge yellow building and that's it. The photo above shows an open-air pool where you can feel comfortable even in winter. It is here where the locals play chess on a "floating" chessboard (the chessboard is not actually floating but placed on a corner of the stone steps that lead to the water). If you are a swimmer you will also enjoy swimming in the 50-meter long open air pool next to this one as the water is comfortably warm even in winter.
The Buda Castle district
Buda Castle (Royal Palace) is on top of Castle Hill and is easily accessible by a funicular train at Adam Clark Square just across the Chain Bridge from the Pest side. The short trip by the funicular train costs 1,000 forint one-way or 1,700 forint for a return ticket (prices as in March 2013). When you get out of the funicular train you will find the Royal Palace on your left (on the right is the annexe to the Royal Palace where the changing of guards ceremony is held). The Royal Palace does not really live up to its name as no royals ever lived in it. It houses the Hungarian National Gallery as well as the Historical Museum of Budapest. As for the annexe to the Royal Palace it is called Sandor Palota in Hungarian and is the official residence and office of the President of the Republic of Hungary. More here.
If you walk further up and continue walking for 100 meters towards your right you will arrive at the must-see tourist spot called the Fisherman's Bastion, passing the colourful Matthias Church (also known as Church of Our Lady) on the way. It stands just in front of the Fisherman's Bastion (singular form for fisherman as that is how it is used in the Hungarian name). You will be able to have a very enjoyable promenade here and take in the breathtaking views of Pest on the other side of the Danube River. The seven turrets there represent the seven Magyar tribes that invaded the Carpathian Basin and established the Principality of Hungary.
Budapest Liszt Ferenc International Airport
It's no hassle at all to get to the city from the Budapest Liszt Ferenc International Airport by taking public transport. As you exit the airport turn right and you will see the bus-stop for Bus No. 200E. It will take you to the Kobanya-Kispest metro terminus of M3 (Line 3). From here you can get off at Deak Ferenc ter if you need to change for M1 or M2. A fourth line (M4) is in its final stages. The airport bus 200E runs every 8-12 minutes in the daytime and every 15-20 minutes in the evening.
The Central Market Hall (Vasarcsarnok)
Another must-see place in Budapest is the Central Market Hall (Nagyvasarcsarnok or simply Vasarcsarnok) at Fovam Square near the Liberty Bridge. In fact Fovam Square is at one end of the famous Vaci utca (utca meaning street). On the top floor are a number of stalls selling typical Hungarian dishes. Not only tourists but many locals too go here to have their meals. And if you are looking for souvenirs to bring home you don't have to go far as this floor is full of souvenir shops that might have just what you are looking for. The market is open from 6 in the morning to 6 in the evening. When you walk out of the market there's a post office just across the road if you have postcards to send.