Private Tour guide for countryside.

I have not yet used him (although I am considering it for this fall), but this one comes recommended.
http://guideinbudapest.fw.hu/Site_Brown/Private_guide_in_Budapest.html

I suggest doing a bit of research on what you want to see, but he has some suggestions on his website.

There are many lovely towns in Hungary, but for day trips you will want to limit the distance covered. One good possibility would be Gyor, which has a very attractive historic center and a bunch of nice, small museums (mostly art-related), in combination with the Pannonhalma Abbey. Although you can make that trip by public transportation (train to Gyor then bus), sometimes stuff happens. It turned out there was a bike race the day of my visit, so the bus route was truncated and I had quite a slog of an uphill walk to the abbey.

Kecskemet is also interesting. You can get there by train, though.

The trio of Szentendre, Visegrad and Esztergom is a popular combo, but I think it's manageable by public transportation (I only did Szentendre), you might to do part of the trip by boat rather than all by car, and I bet there are public tours covering all three towns. I don't know that paying for a private guide would be great value for money compared to the other options.

I trust James E will respond soon. He's the expert on Hungary and certainly can recommend one or more guides as well as suggest which places would most benefit from having a guide and private transportation available.

We would like to see some of the countryside

A1 First tell me your expectation, are you thinking nature or village or town?

A2 Thanks to a rather poor decision involving WWI Hungary lost a lot of the areas of natural beauty (Transylvania) but was left with some pretty extensive and productive farm land ... lots of farm land.

A3 Sure there are a few corners but those are sort of far from Budapest unless you have the time.
A4 But as Carol pointed out there are some very interesting villages and towns (and a few cities like Gyor that still feel like a town) if that interests you.

A5 So the second question, how much time do you have in Budapest and how much of that time do you want to spend out of Budapest, and then maybe I can give you some ideas to check out.

A6 For the guide I like Andrew (Andras) https://www.tripadvisor.com/ShowUserReviews-g274887-d4296354-r524216378-Andras_Illes_Budapest_Tours_Day_Tours-Budapest_Central_Hungary.html and Mail: budapesttours@gmail.com
Phone:+36703654577 Web:https://www.guideinbudapest.freeweb.hu ; But there is an option if you don't want a guide, just want transportation.... Let me know.

We will arrive on a Sunday and leave the following Sunday. We have no scheduled plans. My husband has back issues so walking for long distances may be a problem. We' interested in any suggestions both in the city and out.

A1 Finally, someone spending enough time; then maybe something like this. If you like any of it or have other ideas, let me know and I can help with how to get around. Fortunately most of Budapest is served by trams so there does not need to be a lot of walking and the trams are part of the charm.:

A2 Oh, the guide recommended by TexasTravelmom; I first used him nearly 20 years ago and he is still my go to guy. He has help now and they are good too.

A3
Sunday:
Most flights arrive mid afternoon so a half day in the Buda Castle District.

Monday:
Pest part A. That would be mostly District V, the waterfront, the Parliament, Great Market Hall and the Great Synagogue, Basilica

Tuesday:
Pest part B: That would be Andrassy ut, the Opera House, Heroes’ Square, Szechenyi Baths (great for bad backs)

Wednesday: Pick one
Overnight
Eger (small wine town)
Gyor (old historic district and nearby archabbey)
Pecs (old Ottoman town)
Lillafüred (glorious old “castle” hotel on a lake, nearby forest railway, caves, old castle in Miskolc); better as a two night, but you can cover most in a one night with an early train up and a late train back.
Tihany on Lake Balaton (ancient church, lots of crafts, beautiful views of the lake, nice town to wander)
Vienna (you are that close)

Thursday: returning from above

Friday: Jewish Budapest. Depending on your interest this can be short or long (but had to be completed before sunset for the Sabbath) best done with a guide.

Saturday:
Day trip
Pick one
Szentendre by commuter train and boat trip back to Budapest
Hired guide to Szentendre and either Visegrad or Estergom (trying to do all three is just a lof of work), then the boat ride back
There are some canned trips: Look here and we will see if we can find the vendor (this is a reseller but they aren’t bad) https://www.viator.com/Budapest-tours/Day-Trips-and-Excursions/d499-g5

Sunday: I assume you are boarding the boat?

In the evenings
Restaurant with Gypsy Music
Restaurant on the Danube Corso
Old world Hungarian Restaurant
Crazy restaurant with sword fights and belly dancers (hey, you are on vacation)
The Opera (should be open in September)
The Opperett Theater (also should b open in September)
Really nice Jazz dinner club
Church concert
Walking the corso
Couple of good wine bars
End the trip with an evening trip to the Citadel lookout (Citadella kilátó) Google Maps says its temporarily closed, cant imagine why. Maybe by September. For this you need a taxi, two wine glasses and a bottle of wine. You leave about 8:30pm then sit there and see the most beautiful view in the world (where I fell in love with Budapest nearly 20 years ago) https://upload.wikimedia.org/wikipedia/commons/e/e0/View_from_Gell%C3%A9rt_Hill_by_night._Budapest_%288082258732%29.jpg

Your information has been so helpful and will make my planning so much easier. Do you have recommendations for restaurants and for museums.

B1 For the museums there is so much to choose from, and I am not a big museum hound, so go through this list and if something jumps out, ask and I will tell you if I know anything about it. https://www.offbeatbudapest.com/budapest-city-guide/best-museums/ not on the list, but I like is the Railway museum. https://vasuttortenetipark.hu/en/home/

B2 Restaurants
https://www.rezkakasbistro.hu/ gypsy music
https://sirlancelot.hu/ goofy fun
https://www.mazeltov.hu/ sort of Jewish Hungarian, great space
http://paprikavendeglo.hu/ very Hungarian
https://kiraly100.hu/en/king100restaurant/ old communist party boss hangout, great Hungarian food
https://www.lockerbudapest.com/ perfect breakfast place to be entertained by the family that owns it.
https://www.fricipapa.hu/?lang=en authentic Hungarian dinner sort of place
http://kadarkawinebar.com/ good menu to go with the good wine
https://www.dunacorso.hu/hu/index.php sit out side for the view and the atmosphere
http://spinoza.hu/ with Klezmer concert on Friday (or Saturday, don’t remember)
https://twentysixbudapest.com/ another amazing breakfast place (or dinner, not sure about lunch)
https://pest.vakvarju.com/ sort of relaxed family place with a piano player
https://www.trofea.hu/ all you can eat Hungarian (pretty good too)
http://www.callascafe.hu/ nice atmosphere near the opera (before or after a show) music most nights
https://hachapuri.com/ very good Georgian food
And I bet you will get a few dozen more recommendations. Good food is not in short supply in Budapest.