What Budapest neighborhood best for using public transit

Yay! I think you'll really enjoy it!
The public transportation system is excellent in Budapest so you can easily get wherever you want to go, so don't worry too much about booking a hotel based on accessibility to sites alone. (Of course you may not want to stay where I stayed, way up in the Buda Hills in a residential area:) It was fantastic for the money, but it took quite a while to get to and from. But I arrived without reservations and had to take what I could get:))If I had the chance to go again, I would try to stay in one of the hotels in the Castle Hill area: either on the hill itself, or below Fishermen's Bastion near the Danube. Or, maybe along Andrassy Ut. on the Pest side. It has a "Belle Epoque" feel to it. Some of the hotels in these areas looked pretty swish and expensive, but others looked reasonable.

We stayed near the Great Hall Market when there. Great location for transportation and restaurants. We were there in June a few years ago and the only problem we ran into was no air in the room. That was the summer of the big heat wave in Europe. Budapest can be very humid and hot due to the "frying pan" effect. So main suggestion is to make sure you book a room with air if this bothers you. We actually used a recommended hotel from Rick's book and did love the hotel otherwise.

I just got back from 4 nights in Budapest. I stayed at the Mercure Museum (3-stars). It is about a 20 minute walk to the center of Pest, less than 5 minutes to the metro, trams and buses. I was quite happy to be outside the busy, noisy center, and there were 2 supermarkets, less than 2 blocks away.

You do want to be somewhat central, as the metro and trams stop running around midnight and the taxi drivers will rip you off unmercifully, which is why I wouldn't recommend staying on the Buda side of the river.

The hotel was reasonably priced, very clean and quiet, with a good, full breakfast, and the staff were all very friendly and helpful. The hotel is air-conditioned, but unfortunately the weather while I was there didn't require using it.

After a lot of walking around Castle Hill, go to the Hilton Hotel, next door to Fisherman's Bastion and the church, for a 2-hour wine-tasting experience. The ambience is great and the wine is better. Watch out though, you will probably end up buying a few bottles.

We stayed at the Bellevue B&B on the Buda side. It was maybe a 5-8 minute walk to the Red Metro line 2 at Batthyány tér. Now going back to the B&B was more of an event, walking up hill, and climbing lots of stairs, took maybe 10-15 minutes. It got easier the more we did it and I really found it all to be worth it. We had an amazing balcony view of the Parliament from our B&B. We ate at this wonderful Vegan Hungarian Cafeteria that was steps from the metro. The vibe on the Buda side is way chill. So if you are into working out and burning off some calories, think about the buda side.

If you stay central enough, you won't have to use the public transportation, as Budapest is a "walkable" city. I also think you see/experience more if you walk around. We stayed at the Astoria Residence by Budapesting and we used the transportation only to get to John's Hill as that spot was a bit far from the center.

The public transportation in Budapest is excellent. As one of the other posters said, Budapest is a very walkable city- however in my own experience, I walk so much during the day, I try to use public transportation as much as possible to make travel easier. Usually, I start touring at 9AM and don't stop until after dinner, so if I can cut out some non-essential walking, I do.

Several areas would be prime for you:

Anywhere along Andrassy Ut. is good for the Metro 1 line. This line is barely under the surface, so very easy to access. Pick a hotel near one of the stations, and then transfer at Deak Ter for any of the other lines. Oktogon is a good area on Andrassy with lots of restaurants on Liszt Ferenc Ter. Also, the main tram lines run by Oktogon Ter.

Great Market area has lots of restaurants, and is convenient to the trams, and also the metro. The Danube River is close, which is also nice.

Castle Hill/ Vizivarios neighborhood. Both are atmostpheric, but not very convenient to the trams or metro. If you are looking for convenience, they are not really very central. Plus, Pest has a lot more going on than Buda.

Just my two cents! If you want more, info on Budapest, PM me.