Make plans for two days, one day Buda (west of the Donau) second day Pest (east of Donau)
-take the tram/metro to Moszkva Ter (since 2011, Szell Kalman Ter) or Budapest Deli
-from there you can walk up the mountain and visit the Fisherman's Bastion and the Matthias Church.
-During your walk up you can go via the "hospital in the rock" on the Lovas ut 4. From here there are stairs to the Castle Hill o(Varhegy) to the Fisherman's Bastion.
-Walk in southeasterly direction on the Castle Hill to the Buda Castle. There you have a beautiful view on the Chain Bridge.
-You can walk down or take the cable train down and walk alomg the Donau in southeasterly direction to the Gellert Bath. Entrance cost 5000huf or 15euro
(You can rent everything, towels, bathrobe and you can buy swimming clothing. Perfect for a couple of hours relaxing)
-Take the metro back to the Deli station or Moskvater and climb up the hill (again!!) there are some good restaurants like Pest Buda (google it).
-After dinner when the sun has set, most evenings it's very quiet near the Fisherman's Bastion. Perfect to take pictures of Budapest by night and enjoy the beautiful view.
Trains and metros run till midnight if I remember correctly. about 5 years ago
Second day...the Pest side of the city
-Take the metro to Kossuth lajos ter. Here you can walk around the Parlement (guided tours inside the building have to be booked in advance, they are always full)
-if you walk in southern direction along the boulevard of the Donau you will see the "Shoes on the Danube promenade"
-if you keep walking south you will come across the Chain bridge and on the left Gresham Palace, a beautiful building (and very very expensive hotel)
-there walk in to the Dorottya ut and continue walking to the Vaci ut. This is the main shopping street of Budapest.
-after shopping walk back in the direction of Deak metro station. From there continue to the Andrassy Ut. A very nice boulevard where a lot of (high priced) brands have there exclusive shops.
-if you keep walking on the Andrassy Ut you will find on the ledt side the TerrorHaza (house of terror). A museum that you should not miss when in Budapest (Andrassy Ut 60)
-when outside of the TerrorHaza you can keep walking in northest direction to the end of the Andrassy Ut to Hosok Tere. Here is a monument for the fallen heroes...
-there is a metro station near Hosok Tere which can bring you back to the city for a dinner... about 5 years ago