We visited Budapest in May 2015.
Budapest is a major European city, so there is no dearth of flights into Budapest. The airport itself might leave something to be desired, but that probably depends on the Airlines you are flying with. We took Easyjet and got out of the plane somewhere in the middle of the airport, from where we had to walk like 1km to get inside the airport. But I guess nothing surprises the experienced traveler nowadays - air travel is a chore which needs to be completed somehow to get to the more enjoyable parts of a holiday! Of course, this might be different if you can afford to fly Business class ...
As a tourist, the best place to stay in Budapest is to stay in a hotel close to the Chain Bridge.
The chain bridge is an architectural marvel and often listed as one of the primary attractions of Budapest. So try to book a hotel close to the Chain Bridge. We stayed at the Sofitel (which was earlier Hyatt) and its Bridge view rooms indeed provide a magnificent view of the bridge (e.g. the pic above was taken from our hotel window).
There are actually multiple bridges in sequence connecting Buda and Pest. The primary one is of course also the most majestic - and hence is easy to recognize. For example, check out the Bridge entrance below:
The other attractions e.g. the Parliament (below)
and Fisherman's Bastion (below) are also within walking distance. The Bastion is a very interesting place in its own right. With multiple cafes, lookout points, and the impressive fortified structure - it is easy to spend a whole day in the Bastion.
The whole area around the Chain Bridge looks mesmerizing at night, hence the many night shots. The lights on the Chain Bridge, Parliament House, and Fisherman's Bastion remain ON till around 1am at night - giving it a magical feel as soon as the sun sets.
Another option to explore Budapest at night is to take a cruise. There are multiple cruise lines which cruise the Danube at night. Some also offer the option to have dinner on the boat. We would really recommend taking one of these as they offer the possibility to enjoy most of Budapest's attractions at night without much legwork - which you can then save to enjoy the nightlife :) Below are some pics from our cruise:
The happening spots, e.g. the street Váci utca, which is like a really long street filled with cafes, restaurants, and souvenir shops; are also within walking distance of the Chain Bridge. So it really pays off to stay at a hotel close to the bridge.
If you keep walking along Váci utca (from the Chain Bridge), you will reach the Great Market Hall (below) - which is more like the flea market where you can get souvenirs at bargain prices. You can already see the prices of souvenirs dropping as you get closer to the Great Market.
While Váci utca and the Great Market are good for street shopping, check out Andrassy Avenue for the more trendy and expensive shops e.g. Louis Vuitton, Burberry, and Silka (a local Hungarian fashion brand). A good and easy way to reach Andrassy Avenue is to take the public transport to the Opera (below)
Other places of interest, at least if you are interested in panorama shots of the city (below), include the Gellert Hill lookout points.
Tourism in Hungary is still developing! As a result, it is currently very Budapest centric - there is not much by way of information or organized tours to places outside Budapest - at least for the foreign tourist who does not speak Hungarian.
We tried to search on the interest for inspiration w.r.t. to places to visit outside Budapest. We stumbled upon this article "12 Beautiful photos to make you wonder why you haven't visited Hungary yet" - and I am including the link here as it might be interesting for others as well.
We selected Egerszalok and Bokodi lake - out of which we finalized on Bokodi lake - as Egerszalok was even farther away from Budapest. Although Bokodi lake is closer to Budapest, there is actually an organized tour to Egerszalok - link - so you can check it out if you are interested.
Finally, we booked an private tour with a local driver and set out one fine morning to visit the famed Bokodi-Hutoto lake. Unfortunately, our fears w.r.t. unorganized tourism outside Budapest were well founded, and we did not manage to reach the lake viewpoint depicted in the article. Even our local driver gave up after one hour of trying - it seems that all car roads leading to the lake have been closed - and the only way to reach it is by foot. We only managed to reach a tip of of the lake - so check out our consolation pic of the lake below and let us know if you managed to reach there on your trips.
As with all our tours, we also tried to check out at least one Indian restaurant here. We tried Salaam Bombay which is located just behind the Four Seasons hotel, very close to the Chain Bridge. The restaurant has a good selection of food (menu) items, however the food is just about average. Also, be sure to book in advance as it seems that they have an arrangement with a local Indian tour agent; so the whole restaurant is often booked for Indian tourist groups.