Buda side is possibly the better, I can reccomend a nice hotel a the top of the hills the normafa, for a cheap stay the motelbotel,s are quite nice, we stayed in the Lisa which was a bit south of the center on the pest side, it was a nice room and the food etc was very good it was just a bit far out, the fortuna is a similar and is nearer the city center, however in hot weather air con is a must, having said that the boat on the river with opening windows meant we got a nice cool breeze so it wasnt too bad and for £15 a night we were not complaining.
We also stayed at the Agro Panorama, its dated (1970's) but it was very cheap, and we just had bed and breakfast, possibly the best breakfast, large choice, only trouble is its a long way out and up in the hills easy to get to as the public transport is brilliant, but for a first visit I would get somting more into the center.
Things to see and do.
The Citidel area
The old town and castle area, fishermans bastion, visit the underground caverns in the castle.
Make sure you go up the incline railway up to the castle, its not very long but it is very good.
if you have any interest in it the police musem is interesting,
Hero square, and then into the park, the transport museum is fantastic (small but fantastic) if the weather is good then you have to visit one of the outdoor swiming pools or spas, for swiming and a hot pool I would recommend the one on Margret Island not on the end of the island but half way down, it has a warter park and hot spa pool, for a spa then Gellert Baths
Opposite Gellert Baths (same side of the river) little church built underground in the hill is a nice little 20min visit.
Then cross over the brigde and visit the indoor market just the other side of the river.
Do the Tourist thing and walk up Vaci Ute, loads of outdoor restraunts and shops to look at.
A bit somber but the house of terror, is a building used by secret police before during and after the war, now an exibition and well worth a look, as is the Jewish musuem however its not feel good visit but it is educational.
The Budapest musuem is very good, starts at pre history in basement and as you go up the building you go up through the ages.
Very well worth looking at are the roman remains just outside Budapest at Aquincum, really good they have these view screens that look like seeside telescopes, and when you view the site through them it gives a computer generated view of how the town looked. You get there on a local train took about 20 minutes from the station at Margrit Island bridge, nice trip as the train goes along the river.
For a bit of a laugh visit the Sir Lancelot restraunt, its a bit artificial but the food is pretty good and its done well.
Transport, FIRST THING go to any of the big squares where the trams stop Boraras Ter/Moszkva tér or one of the others, buy a tranport map, it has all the bus routes by bus number, all the trams by number and the trains, and the underground.
If you are going to visit attractions, then a tourist travell card is good value as it give free or reduced price on visits, plus you can use it on all local trains/trams/underground/bus routes.
If you dont think you will vist attractions then just get a 3 or 5 day travel card which cheaper.
Well there is my info if you want any more get back to me, I have been once or twice. I love it to bits, nice people, good food, great local beer.
Though that intrepid explorer Ting Tong Tom managed to stay for one day and see the sites, without help. 
